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IN  MEMORIAM 
FLORIAN  CAJORI 


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WELLS'   MATHEMATICAL  SERIES. 


Academic  Algebra. 

Higher  Algebra. 

University  Algebra. 

College  Algebra. 

Plane  Geometry. 

Solid  Geometry. 

Plane  and  Solid  Geometry. 

Plane  and  Spherical  Trigonometry. 

Plane  Trigonometry. 

Essentials  of  Trigonometry. 

Logarithms  (flexible  covers). 

Elementary  Treatise  on  Logarithms. 


Special  Catalogue  and  Terms  ok 
application. 


SHORT   COURSE 


IN 


HIGHEE  ALGEBKA 


ACADEMIES,  HIGH  SCHOOLS,  AND  COLLEGES. 


BT 


WEBSTER  WELLS,  S.B., 

ASSOCIATE  PROFESSOR  OF  MATHEMATICS  IN  THE  UA88ACHUSKTTS 
INSTITUTE  OF  TECHNOLOGY. 


LEACH,   SHEWELL,    &   SANBORN, 
BOSTON  AND  NEW  YORK. 


COPTRIGHT,  1889, 

Bt  WEBSTER  WELLS. 


J.  S.  CiTSHiNQ  &  Co.,  Printers,  Boston. 


2.^ 


PREFACE. 


THIS  volume  has  been  prepared  in  response  to  a  demand 
from  numerous  teachers  for  a  work  intermediate  in 
its  scope  between  the  author's  Academic  and  University 
Algebras.  The  first  24f  pages  of  the  Higher  Algebra  are 
the  same  as  the  corresponding  pages  of  the  Academic  Al- 
gebra. The  work  on  pages  249  to  305  of  the  latter  has  been 
rewritten  with  reference  to  the  new  matter,  and  71  pages 
have  been  added  to  the  book ;  and  it  is  now  put  forth  as  a 
complete  preparatory  text,  containing  all  the  topics  required 
for  admission  to  any  of  the  Colleges,  Universities,  or  Scien- 
tific Schools  of  the  country. 

The  new  matter  is  contained  principally  in  the  following 

chapters : 

\ 
XXVI.    Inequalities. 

XXVII.    The  Theory  of  Limits;    Interpretation  of  the 

-  a     a         -,0 

forms  -,  — ,  and  — 

0    ^  0 

XXIX.    Variation. 
XXXII.    Harmonical  Progression. 
XXXIV.    The  Theorem  of  Undetermined  Coefficients. 
XXXV.    The  Binomial  Theorem  ;  Fractional  and  Nega- 
tive Exponents. 
XXXVII.    Compound  Interest  and  Annuities. 
XXXVIII.    Permutations  and  Combinations. 
XXXIX.    Continued  Fractions. 

There  is  also  given  in  connection  with  the  chapter  on 
Logarithms,  "a  discussion  of  Lo^garithmic  and  Exponential 
Series. 


!Oi:?OR036 


iv  PREFACE. 

Attention  is  respectfully  invited  to  the  following  among 
the  many  new  features  of  the  book : 

1.  The  method  of  factoring  quadratic  expressions,  Art. 
283  ;  and  the  examples  in  the  same  article  illustrating  the 
factoring  of  expressions  of  six  terms. 

2.  The  method  of  interpreting  the  forms  -  and  ~  ;  Arts. 
300  and  301.  ^ 

3.  The  fundamental  ideas  with  regard  to  convergency  and 
divergency  of  series  ;  Arts.  371  to  373. 

4.  The  method  given  in  Ex.  2,  Art.  381,  for  finding  the 
coefficients  when  separating  a  fraction  into  its  partial  frac- 
tions. 

5.  The  proof  of  the  Binomial  Theorem  for  any  form  of 
exponent,  Arts.  389  to  391  ;  especially  the  general  proof,  in 
Art.  390,  of  the  law  of  formation  of  the  successive  coeffi- 
cients. 

6.  The  proof  of  the  formula  for  the  number  of  permuta- 
tions of  n  quantities  taken  r  at  a  time  ;  Art.  447. 

WEBSTER   WELLS. 


COl^TEI^TS. 


TAQM 

I.    Definitions  and  Notation 1 

Symbols  of  Quantity 1 

Symbols  of  Operation 2 

Symbols  of  Relation 4 

Symbols  of  Abbreviation 4 

Algebraic  Expressions 5 

Axioms 10 

Solution  of  Problems  by  Algebraic  Methods  ...  10 

Negative  Quantities 13 

n.    Addition 15 

Addition  of  Similar  Terms 16 

Addition  of  Polynomials 18 

III.  Subtraction 20 

Subtraction  of  Polynomials 21 

IV.  Use  of  Parentheses 24 

V.    Multiplication 27 

Multiplication  of  Monomials 29 

Multiplication  of  Polynomials  by  Monomials      .     .  30 

Multiplication  of  Polynomials  by  Polynomials    .     .  31 

VI.     Division 36 

Division  of  Monomials 37 

Division  of  Polynomials  by  Monomials 38 

Division  of  Polynomials  by  Polynomials    ....  39 

VII.     FORMUL.E 44 

VIIL     Factoring 51 


VI  conte:n^ts. 

PAGE 

IX.    Highest  Common  Factor 65 


X.    Lowest  Common  Multiple 75 

XL    Fractions 80 

General  Principles 80 

To  Reduce  a  Fraction  to  its  Lowest  Terms     ...  82 
To  Reduce  a  Fraction  to  an  Entire  or  Mixed  Quan- 
tity      86 

To  Reduce  a  Mixed  Quantity  to  a  Fractional  Form  88 
To   Reduce   Fractions   to    their   Lowest   Common 

Denominator 89 

Addition  and  Subtraction  of  Fractions 91 

Multiplication  of  Fractions 97 

Division  of  Fractions 99 

Complex  Fractions 101 

XII.     Simple  Equations 106 

Transposition 107 

Solution  of  Simple  Equations 108 

Solution  of  Equations  containing  Fractions    .     .     .  Ill 

Solution  of  Literal  Equations 115 

Solution  of  Equations  involving  Decimals  ....  117 

XIII.  Problems    leading   to    Simple    Equations   con- 

taining One  Unknown  Quantity   ....  119 

XIV.  Simple    Equations    containing    Two    Unknown 

Quantities 132 

Elimination  by  Addition  or  Subtraction     .     .     .     .  133 

Elimination  by  Substitution 135 

Elimination  by  Comparison 136 

XV.     Simple   Equations   containing  more  than  Two 

Unknown  Quantities 144 

XVI.  Problems  leading  to  Simple  Equations  con- 
taining more  than  One  Unknown  Quan- 
tity    148 


CONTENTS.  vii 

PAGE 

XVII.     Involution 158 

Involution  of  Monomials 158 

Square  of  a  Polynomial 159 

Cube  of  a  Binomial 161 

Any  Power  of  a  Binomial 162 

XVIII.     Evolution 165 

Evolution  of  Monomials 165 

Square  Root  of  Polynomials 167 

Square  Root  of  Numbers .  170 

Cube  Root  of  Polynomials 173 

Cube  Root  of  Numbers 176 

XIX.    The  Theory  of  Exponents 180 

XX.    Radicals 190 

To  Reduce  a  Radical  to  its  Simplest  Form    .     .  190 

Addition  and  Subtraction  of  Radicals  ....  193 
To   Reduce   Radicals   of   Different   Degrees  to 

Equivalent  Radicals  of  the  Same  Degree    .     .  195 

Multiplication  of  Radicals 196 

Division  of  Radicals 198 

Involution  and  Evolution  of  Radicals   ....  199 
To  Reduce  a  Fraction  having  an  Irrational  De- 
nominator to  an  Equivalent  Fraction  whose 

Denominator  is  Rational 200 

Imaginary  Quantities 203 

Multiplication  of  Imaginary  Quantities     .     .     .  204 

Properties  of  Quadratic  Surds 206 

Square  Root  of  a  Binomial  Surd 207 

Solution  of  Equations  containing  Radicals    .     .  208 

XXI.    Quadratic  Equations 211 

Pure  Quadratic  Equations 211 

Affected  Quadratic  Equations 212 

First  Method  of  Completing  the  Square    .     .     .  213 

Second  Method  of  Completing  the  Square     .     .  216 

Solution  of  Quadratic  Equations  by  a  Formula .  222 

XXII.     Problems  involving  Quadratic  Equations.  223 


viii  COI^TENTS. 

PAGlS 

XXIII.    Equations  in  the  Quadratic  Form  ....  228 

XXIV*    Simultaneous    Equations    involving    Quad- 
ratics     233 

XXV.   Theory  of  Quadratic  Equations      ....  246 

Factoring 250 

Discussion  of  the  General  Equation      ....  254 

XXVI.   Inequalities      .     « 256 

XXVII.   The  Theory  op  Limits 261 

Interpretation  of  - 262 

Interpretation  of  — 263 

The  Problem  of  the  Couriers 263 

XXVIII.   Ratio  and  Proportion 267 

Properties  of  Proportions 268 

XXIX.   Variation « 276 

XXX.   Arithmetical  Progression 280 

XXXI.    Geometrical  Progression 289 

XXXII.   Harmonical  Progression 298 

XXXIII.  The  Binomial  Theorem;   Positive  Integral 

Exponent 301 

XXXIV.  The  Theorem  of  Undetermined  Coefficients,  306 

The  Theorem  of  Undetermined  Coefficients  .  .  308 
Application  to  the  Expansion  of  Fractions  into 

Series 310 

Application  to  the  Expansion  of  Radicals  into 

Series 313 

Application  to  the  Decomposition  of   Rational 

Fractions 314 

Application  to  the  Reversion  of  Series  ....  320 


CONTENTS.  IX 

PAGB 

XXXV.   The  Binomial  Theorem;  Fractional  and 

Negative  Exponents 323 

XXXVI.   Logarithms 330 

Properties  of  Logarithms 332 

Use  of  the  Table 340 

Applications 343 

Arithmetical  Complement 345 

Exponential  Equations 348 

Exponential  and  Logarithmic  Series  ....  350 

XXXVII.   Compound  Interest  and  Annuities  ....  354 

Annuities 357 

XXXVIII.   Permutations  and  Combinations    ....  360 

XXXIX.   Continued  Fractions 366 

Properties  of  Convergents 370 

Answers. 


ALGEBRA. 


I.   DEFINITIONS  AND  NOTATION. 

1.  Algebra  is  that  branch  of  mathematics  in  which  the 
relations  of  numbers  are  investigated,  and  the  reasoning 
abridged  and  generalized  by  means  of  symbols. 

Note.  Writers  on  Algebra  employ  the  word  "  quantity  "  as  synony- 
mous with  "  number " ;  this  definition  of  the  word  will  be  understood 
throughout  the  present  work. 

2.  The  Symbols  of  Algebra  are  of  four  kinds  : 

1.  Symbols  of  Quantity. 

2.  Symbols  of  Operation. 

3.  Symbols  of  Relation. 

4.  Symbols  of  Abbreviation. 

SYMBOLS   OF   QUANTITY. 

3.  The  symbols  of  quantity  generally  used  are  the  figures 
df  Arithmetic,  and  the  lettei's  of  the  alphabet. 

Figures  are  used  to  represent  known  quantities  and  deter- 
mined values ;  while  letters  may  represent  any  quantities 
whatever,  known  or  unknown. 

4.  Known  Quantities,  or  those  whose  values  are  given, 
when  not  expressed  b}"  figures,  are  usually  represented  by 
the  first  letters  of  the  alphabet,  as  a,  6,  c. 

5.  Unknown  Quantities,  or  those  whose  values  are  to  be 
determined,  are  usually  represented  by  the  last  letters  of  the 
alphabet,  as  x,  y^  z. 


2-  ALGEBRA. 

6.  Quantities  occupying  similar  relations  in  the  same 
problem,  are  often  represented  by  the  same  letter,  distin- 
guished b}"  different  accents;  as  a',  a",  a'",  read  "  a  prime," 
"  a  second,"  "  a  third,"  etc. 

They  may  also  be  distinguished  by  different  subscript 
figures;  as  «!,  as,  ag,  read  "a  one,"  "a  two,"  "a  three," 
etc. 

7.  Zero,  or  the  absence  of  quantity,  is  represented  by  the 
symbol  0. 

SYMBOLS   OF  OPERATION. 

8.  The  Sign  of  Addition,  +,  is  called  ^^ plus.'' 

Thus,  a +  6,  read  "a  plus  6,"  indicates  that  the  quantity 
h  is  to  be  added  to  the  quantity  a. 

9.  The  Sign  of  Subtraction,  — ,  is  called  "  minus.'' 
Thus,  a  —  b,  read  "a  minus  b,"  indicates  that  the  quan- 
tity b  is  to  be  subtracted  from  the  quantity  a. 

Note.  The  sign  '^  indicates  the  difference  of  two  quantities ;  thus, 
a^^  b  denotes  that  the  difference  of  the  quantities  a  and  b  is  to  be 
found. 

10.  The  Sign  of  Multiplication,  x,  is  read  '^  times," 
"  into,''  or  "  multiplied  by." 

Thus,  a  X  b  indicates  that  the  quantity  a  is  to  be  multi- 
plied by  the  quantity  b. 

The  sign  of  multiplication  is  usually  omitted  in  Algebra, 
except  between  arithmetical  figures ;  the  multiplication  of 
quantities  is  therefore  indicated  by  the  absence  of  any  sign 
between  them.     Thus,  2ab  indicates  the  same  as  2  x  a  X  &. 

A  point  is  sometimes  used  in  place  of  the  sign  x  between 
two  or  more  figures  ;  thus,  2  •  3  •  4  denotes  2x3x4. 

11.  Quantities  multiplied  together  are  called  factors,  and 
the  result  of  the  multiplication  is  called  the  product. 

Thus,  2,  a,  and  b  are  the  factors  of  the  product  2ab, 


DEFINITIONS  AND  NOTATION.  3 

12.  A  Coefficient  is  a  number  prefixed  to  a  quantity  to 
indicate  how  man}^  times  the  quantity  is  to  be  taken. 

Thus,  in  Aax,  4  is  the  coefficient  of  ax,  and  indicates  that 
ax  is  to  be  taken  4  times;  that  is,  4: ax  is  equivalent  to 
ax -\- ax -j- ax -\-  ax. 

When  no  coefficient  is  expressed,  1  is  understood  to  be 
the  coefficient.     Thus,  a  is  the  same  as  1  a. 

When  any  number  of  factors  are  multiplied  together,  the 
product  of  any  of  them  ma}'  be  regarded  as  the  coefficient 
of  the  product  of  the  others.  Thus,  in  abed,  ah  is  the 
coefficient  of  cc? ;  b  of  acd  ;  abd  of  c  ;  etc. 

13.  An  Exponent  is  a  figure  or  letter  written  at  the  right 
of,  and  above  a  quantity',  to  indicate  the  number  of  times 
the  quantity  is  taken  as  a  factor. 

Thus,  in  a.-^,  the  ^  indicates  that  x  is  taken  three  times  as  a 
factor  ;  that  is,  y?  is  equivalent  to  xxx. 

14.  The  product  obtained  by  taking  a  factor  two  or  more 
times  is  called  a  power.  A  single  letter  is  also  often  called 
i]iQ  first  power  of  that  letter.     Thus, 

a^  is  read  "  a  to  the  second  power,"  or  "  a  square,*^  and 
indicates  aa ; 

a^  is  read  "  a  to  the  third  power,"  or  "  a  cube,'"  and  indi- 
cates aaa; 

a*  is  read  "  a  to  the  fourth  power,"  or  '•'•  a  fourth"  and 
indicates  aaaxi ;  etc. 

When  no  exponent  is  written,  the  first  power  is  under- 
stood ;  thus,  a  is  the  same  as  a^. 

15.  The  Sign  of  Division,  -^,  is  read  "  divided  by." 
Thus,  a-i-b  denotes  that  the  quantity  a  is  to  be  divided 

by  the  quantity  b. 

Division  is  also  indicated  by  writing  the  dividend  above, 

and  the  divisor  below,  a  horizontal  line.     Thus,  -  indicates 

a  ^ 

the  same  as  a -5-  6.     When  thus  written,  -  is  often  read  "  a 

over  b." 


ALGEBRA. 


SYMBOLS   OF  RELATION. 


16.  The  symbols  of  relation  are  signs  used  to  indicate 
the  relative  magnitudes  of  quantities. 

17.  The  Sign  of  Equality,  =,  read  ^^  equals,''  or  "is 
equal  to,''  indicates  that  the  quantities  between  which  it  is 
placed  are  equal. 

Thus,  x  =  y  indicates  that  the  quantities  x  and  y  are  equal. 

A  statement  that  two  quantities  are  equal  is  called  an 
equatio7i. 

Thus,  a;+4=2a;— lisan  equation,  and  is  read  "  x  plus 
4  equals  2x  minus  1." 

18.  The  Sign  of  Inequality,  >  or  < ,  read  ' '  is  greater 
than  "  and  "  is  less  tha7i "  respectively,  when  placed  between 
two  quantities,  indicates  that  the  quantity  toward  which  the 
opening  of  the  sign  turns  is  the  greater. 

Thus,  x>y  is  read  "  ic  is  greater  than  y"  ;  x  —  6<y  is 
read  "  a?  minus  6  is  less  than  y," 


SYMBOLS  OF  ABBREVIATION. 

19.  The  Sign  of  Deduction,  .'.,  stands  for  therefore  or 
hence. 

20.  The  Signs  of   Aggregation,  the  parenthesis   ( ) ,  the 

brackets  [  ] ,  the  braces  \  \ ,  and  the  vinculum ,  indicate 

that  the  quantities  enclosed  by  them  are  to  be  taken  col- 
lectively.    Thus, 


(«  +  &)»,    [_a-\-b']x,    \a-\-b\x,    a  +  bxx, 

all  indicate  that  the  quantity  obtained  by  adding  a  and  b  is 
to  be  multiplied  by  x. 


DEFINITIONS   AND  NOTATION.  5 

21.  The  Sign  of  Continuation,  ...,  stands  for  "and  so 
on  "  or  "  continued  by  the  sayne  law.''     Thus, 

a,  a  + 5,  a +  26,  a  4- 3  6,  ... 

reads  "a,  a  plus  6,  a  plus  26,  a  plus  3  6,  and  so  on." 

ALGEBRAIC  EXPRESSIONS. 

22.  An  Algebraic  Expression  is  any  combination  of  alge- 
braic symbols  ;  as  2  a^  —  3  a6  -f  c^. 

23.  A  Term  is  an  algebraic  expression  whose  parts  are  not 

separated  by  the  signs  +  or  —  ;  as  2ar,  —  3a6,  or  -f  &. 

2a?^  —Sab,  and  c^  are  called  the  terms  of  the  expression 

2a^-3a6  +  c^. 

24.  Positive  Terms  are  those  preceded  by  a  plus  sign  ;  as 

+  2ic2,  or  +c». 

For  this  reason,  the  sign  +  is  often  called  the  positive 
sign.  If  no  sign  is  expressed,  the  term  is  understood  to  be 
positive  ;  thus,  a  is  the  same  as  +a. 

25.  Negative  Terms  are  those  preceded  by  a  minus  sign ; 
as 

—  3  ab,  or  —  bc^. 

For  this  reason,  the  sign  —  is  often  called  the  negative 
sign;  it  can  never  be  omitted  before  a  negative  term. 

Note.  In  a  negative  term,  the  numerical  coefficient  indicates  how 
many  times  the  quantity  is  to  be  taken  suhtractivehj .  (Compare  Art. 
12.) 

Thus,  —  3a6  is  equivalent  to  —ah  —  ab  —  ah. 

26.  In  Arithmetic,  if  the  same  number  be  both  added  to 
and  subtracted  from  another,  the  value  of  the  latter  will  not 
be  changed.     Thus, 

5  +  3-3  =  5. 


e  ALGEBRA. 

Similarly,  in  Algebra,  if  any  quantity  h  be  both  added  to 
and  subtracted  from  another  quantity  a,  the  result  will  be 
equal  to  a.     That  is, 

a-\-h  —  h  —  a. 

Consequently,  equal  terms  affected  by  unlike  signs,  in  an 
expression,  neutralize  each  other,  or  cancel. 

--.      27.   A  Monomial  is  an  algebraic  expression  consisting  of 
only  one  term  ;  as  5  a,  7  a6,  or  —  3  hh. 

A  monomial  is  sometimes  called  a  simple  quantity. 

__^__^         28.   A  Polynomial  is  an  algebraic  expression  consisting  of 
more  than  one  term  ;  as  a  +  6,  or  3  a^  +  6  —  5  6^. 

A  polynomial  is  sometimes  called  a  compound  quantity. 

29.  A  Binomial  is  a  polj^nomial  of  two  terms ;  as  a  —  b, 
or  2a +  62. 

30.  A   Trinomial   is   a  polynomial   of    three   terms ;    as 

— ~       31.   Similar  or  Like  Terms  are  those  which  differ  only  in 
their  numerical  coefficients.     Thus, 

2xy^  and  —  7xi/  are  similar  terms. 

—      32.   Dissimilar  or  Unlike  Terms  are  those  which  are  not 
similar.     Thus, 

boi^y  and  bxy^  are  dissimilar  terms. 

33.  The  Degree  of  a  term  is  the  number  of  literal  factors 
which  it  contains.     Thus, 

2  a  is  of  the  Jirst  degree,  since  it  contains  but  07ie  literal 
factor ; 

ab  is  of  the  second  degree,  since  it  contains  two  literal 
factors ; 

da^b^  is  of  the  fifth  degree,  since  it  contains  Jive  literal 
factors. 


DEFINITIONS  AND  NOTATION.  T 

The  degree  of  any  term  is  determined  by  adding  the  expo- 
nents of  its  several  letters.     Thus, 

a6V  is  of  the  sixth  degree. 

34.  Homogeneous  Terms  are  those  of  the  same  degree. 
'       a^,  35c,  and  —  4ar  are  homogeneous  terms. 

35.  A  polynomial  is  called  homogeneous  when  all  its  terms 
are  homogeneous  ;  as  a^  +  2  ahc  —  3  h^. 

36.  A  polynomial  is  said  to  be  arranged  according  to  the 
ascending  powers  of  any  letter,  when  the  term  containing 
the  lowest  exponent  of  that  letter  is  placed  first,  that  having 
the  next  higher  immediately  after,  and  so  on.     Thus, 

h^ -f  3a6« -  2a262  j^^a%-4.a^ 

is  arranged  according  to  the  ascending  ix)wers  of  a. 

Note.  The  term  6*,  which  does  not  involve  a  at  all,  is  regarded  as 
containing  the  lowest  exponent  of  a  in  the  above  expression. 

37.  A  polynomial  is  said  to  be  arranged  according  to  the 
descending  powers  of  any  letter,  when  the  term  containing 
the  highest  exponent  of  that  letter  is  placed  first,  that  having 
the  next  lower  immediately  after,  and  so  on.     Thus, 

is  arranged  according  to  the  descending  powers  of  h. 

38.  The  Reciprocal  of  a  quantity  is  1   divided  by  that 

quantity.     Thus,  the  reciprocal  of  a  is  _ ;  and  of  x-{-y  is 
1  a 


x-\-y 

39.  The  Numerical  Value  of  an  expression  is  the  result 
obtained  by  rendering  it  into  an  arithmetical  quantity,  by 
means  of  the  numerical  values  assigned  to  its  letters. 


8  ALGEBRA. 

Thus,  the  numerical  value  of 

when  a  =  4,  5  =  3,  c  =  5,  and  d  =  2,  is 

4  X  4  +  3  X  3  X  5  -  23  =  16  +  45  -  8  =  53. 

EXERCISES. 

40.  Find  the  numerical  value  of  the  following  expressions, 
when  a  =2,  5  =  3,  c=l,  and  d  =  4  : 

1.  a2  +  2a6-c  +  d.  6.    -- 

a"* 

2.  Sa^--2a-b-{-cK  v     cd      ah 


'■^-^ 


3.  5 a^b-^- 4. ab^— 27 cd.  8.    ¥  -  a^bK 

4.  2a2  +  36c-l.  9.   ^' -^. 

cd  5  ac      3  6^ 

&^c^d  b'^c'^d' 

If  the  expression  involves  parentheses,  the  operations  indi- 
cated within  the  parentheses  must  be  performed  ^rs^. 

Thus,  to  find  the  numerical  value,  when  a  =  3,  6=2,  and 

b(2a  -  3c)  {a?  +  c^)  -  V^,' 
^  a-  —  b- 

we  havt, 

2a-3c=6-3=    3 

a2  +  c2  =  9  +  l  =  10 

a2+62  =  9_p4  =  13 

a2-62=9_4=:    5 

Hence  the  numerical  value  of  the  expression  is 

2x3xlO-^  =  60-^  =  M. 
5  5         5 


DEFINITIONS  AND  NOTATION.  9 

Find  the  numerical  values  of  the  following,  when  a  =  4, 
5  =  2,  c  =  3,  and  d=  1  : 

11.  a\a-i-b)-2abc,  iq  ^        ]  ^. 

3a  — 3c      3 

12.  7a'  +  {a-b){a-c), 

.tf     25a  — 30c  — d 

13.  15a-7(6  +  c)-d.  ^^-   ^"ip"^ 

14.  cCa^+^'  +  a'-O.  ^g     a2^52     ^,52 

15.  25a2_7(52  +  c2)  +  d2.  *    «'-^'      ^  +  ^' 

Find  the  values  of  the  following,  when  a  =  -,  Z>  =  -,  c  =  -, 
and  a;  =  2  : 

19.    (2a +  35  + 5c)  (8a +  36 -5c)  (2a -36  + 15c). 
21.    a:*-(2a  +  36)a.-3  +  (3a-26)ar^-ca;  +  5c. 


22. 


6 

2  a; 


8  6c  —  a     a  +  6  +  c 


41.  Put  the  following  into  the  form  of  algebraic  expres- 
sions : 

1.  Five  times  a,  added  to  twice  6. 

2.  Two  times  a;,  minus  y  to  the  second  power. 

3.  The  product  of  a,  6,  c  square,  and  d  cube. 

4.  Three  times  the  cube  of  a,  minus  twice  the  product  of 
a  square  and  6,  plus  the  cube  of  c. 

6.  The  product  of  x  +  y  and  a. 

6.  The  product  ot  x -^  y  and  a  —  6. 

7.  a  square,  divided  by  the  product  of  6  and  c. 

8.  a  square,  divided  by  6  —  c. 


10  ALGEBRA. 

9.    X  divided  by  3,  plus  2,  equals  three  times  y  minus  11. 

10.  The  product  of  m  and  a  -f  6  is  less  than  the  recipro- 
cal of  X  cube. 

AXIOMS. 

42.  An  Axiom  is  a  truth  assumed  as  self-evident. 

Algebraic  operations  are  based  upon  definitions,  and  the 
following  axioms : 

1.  If  equal  quantities  be  added  to  equal  quantities,  the 
sums  will  be  equal. 

2.  If  equal  quantities  be  subtracted  from  equal  quanti- 
ties, the  remainders  will  be  equal. 

3.  If  equal  quantities  be  multiplied  by  equal  quantities, 
the  products  will  be  equal. 

4.  If  equal  quantities  be  divided  by  equal  quantities,  the 
quotients  will  be  equal. 

5.  If  the  same  quantity  be  both  added  to  and  subtracted 
from  another,  the  value  of  the  latter  will  not  be  changed. 

6.  If  a  quantity  be  both  multiplied  and  divided  by 
another,  the  value  of  the  former  will  not  be  changed. 

7.  Quantities  which  are  equal  to  the  same  quantity  are 
equal  to  each  other. 

SOLUTION  OF  PROBLEMS  BY  ALGEBRAIC  METHODS. 

43.  The  following  examples  will  illustrate  the  application 
of  the  notation  of  Algebra  in  tlie  solution  of  problems. 

1.  The  sum  of  two  numbers  is  30,  and  the  greater  is  4 
times  the  less.     What  are  the  numbers? 

AVe  will  first  solve  the  problem  by  the  method  of  Arith- 
metic, and  afterwards  by  Algebra.  The  marginal  letters 
refer  to  the  corresponding  steps  of  the  two  methods ;  that  is, 
the  operation  (a)  in  the  algebraic  solution  is  equivalent  to 
the  operation  (a)  in  the  arithmetical ;  and  so  on.  In  this 
way  the  student  can  compare  the  two  processes  step  by  step. 


DEFINITIONS  AND  NOTATION.  H 


Solution  by  Akithmetic. 

The  less  number,  plus  the  greater  numher,  equals  30. 

(a)  Hence  the  less  number,  plus  4  times  the  less  number,  equals  30 

(b)  Therefore  5  times  the  less  number  equals  30. 

(c)  Hence  the  less  number  is  one-fifth  of  30,  or  6. 
(rf)  Then  the  greater  number  is  4  times  6,  or  24. 

Solution  by  Algebra. 

Let  X  =  the  less  number. 

Then  4  or  =  the  greater  number. 

(a)  By  the  conditions,  x+4x  =  30. 

(b)  Or,  6x=  30. 

(c)  Dividing  by  5,  x  =    6,  the  less  number. 
{d)  Whence,                              4a:  =  24,  the  greater  number. 

2.  A,  B,  and  C  together  have  $GG.  A  has  one-half  as 
much  as  B,  and  C  has  as  much  as  A  and  B  together.  How 
much  has  each  ? 

Let  X  =  the  number  of  dollars  A  has. 

Then  2  x  =  the  number  of  dollars  B  has, 
and                              x  +  2x,  or  3 a:  =  the  number  of  dollars  C  has. 

By  the  conditions,     x+2x-{-Sx  =  66. 

Or,  6x  =  66. 

Whence,  x  =  11;  the  number  of  dollars  A  has. 

Therefore,  2  a:  =  22,  the  number  of  dollars  B  has, 

and  3a:  =  33,  the  number  of  dollars  C  has. 

3.  The  sum  of  the  ages  of  A  and  B  is  109  years,  and  A  is 
13  years  younger  than  B.     What  are  their  ages? 

Let  X  =  the  number  of  years  in  A's  age. 

Then  a:  +  13  =  the  number  of  years  in  B*s  age. 

By  the  conditions,        a:  +  a:  +  13  =  109. 

Or,  2x-|-13=109. 

Whence,  2  a:  =96. 

And,  a:  =  48,  the  number  of  years  in  A's  age 

Therefore,  a:  +  13  =  61,  the  number  of  years  in  B'a  agfr 


12  ALGEBRA. 


PROBLEMS. 


4.  The  greater  of  two  numbers  is  5  times  the  less,  and 
their  sum  is  42.     What  are  the  numbers  ? 

5.  The  sum  of  the  ages  of  A  and  B  is  68  years,  and  B  is 
6  years  older  than  A.     What  are  their  ages  ? 

6.  Divide  $1200  between  A  and  B,  so  that  A  may  receive 
$128  less  than  B. 

7.  A  man  had  $3.72  ;  after  spendiug  a  certain  sum,  he 
found  that  he  had  left  3  times  as  much  as  he  had  spent. 
How  much  had  he  spent? 

8.  Divide  $260  between  A,  B,  and  C,  so  that  B  may 
i*eceive  3  times  as  much  as  A,  and  C  3  times  as  much  as  B. 

9.  Divide  the  number  125  into  two  parts,  one  of  which  is 
21  less  than  the  other. 

10.  The  sum  of  three  numbers  is  98  j  the  second  is  3 
times  the  first,  and  the  third  exceeds  the  second  by  7. 
What  are  the  numbers? 

11.  A,  B,  and  C  together  have  $127 ;  C  has  twice  as 
much  as  A,  and  $13  more  than  B.     How  much  has  each? 

12.  My  horse,  carriage,  and  harness  together  are  worth 
$400.  The  horse  is  worth  11  times  as  much  as  the  harness, 
and  the  carriage  is  worth  $175  less  than  the  horse.  What 
is  the  value  of  each? 

13.  The  sum  of  three  numbers  is  108.  The  first  is  one- 
third  of  the  second,  and  33  less  than  the  third.  What  are 
the  numbers? 

14.  Divide  the  number  210  into  three  parts,  such  that  the 
first  is  one-half  of  the  second,  and  one-third  of  the  third. 

15.  A  man  bought  a  cow,  a  sheep,  and  a  hog,  for  $75  ; 
the  price  of  the  sheep  was  $27  less  than  the  price  of  the 
cow,  and  $6  more  than  the  price  of  the  hog.  What  was  the 
price  of  each? 


DEFINITIONS  AND  NOTATION.  13 


NEGATIVE  QUANTITIES. 

44.  The  signs  -f  and  — ,  besides  indicating  the  opera- 
tions of  addition  and  subtraction,  are  also  used,  in  Algebra, 
to  distinguish  between  quantities  which  are  the  exact  reverse 
of  each  other  in  quality  or  condition. 

Thus,  in  the  thermometer,  we  may  speak  of  a  temperature 
above  zero  as  -f,  and  of  one  below  as  — .  For  example, 
+  25°  means  25°  above  zero,  and  — 10°  means  10°  below 
zero. 

In  navigation,  north  latitude  is  considered  -f ,  and  south 
latitude  —  ;  west  longitude  is  considered  -{-,  and  east  lon- 
gitude — .  For  example,  a  place  in  latitude  —30°,  longi- 
tude +  95°,  would  be  in  latitude  30°  south  of  the  equator, 
and  in  longitude  95°  west  of  Greenwich. 

Again,  in  financial  transactions,  we  may  consider  assets 
as  +?  £^nd  debts  or  liabilities  as  — .  For  example,  the 
statement  that  a  man's  property  is  —  $100,  means  that  he 
owes  or  is  in  debt  $100. 

And  in  general,  when  we  have  to  consider  quantities  the 
exact  reverse  of  each  other  in  quality  or  condition,  we  may 
regard  quantities  of  either  quality  or  condition  as  positive, 
and  those  of  the  opposite  quality  or  condition  as  negative. 

45.  The  thermometer  affords  an  excellent  illustration  of 
the  relation  between  positive  and  negative  quan- 

i  Q  titles. 

4-  5  Let    OA  represent  the   scale  for  temperatures 

[j^g  above  zero,   and  OB  the   scale   for  temperatures 

+  2  below  zero ;    and   let   us   consider   the   foUowinoj 


0 


B 


"^  Q  problem : 

—  1  At  7  A.M.  the  temperature  is  —  6°  ;  at  noon  it  is 
~g  11°  warmer,  and  at  6  p.m.  it  is  9°  colder  than  at 

—  4  noon.     Required  the  temperatures  at  noon  and  at 
2q  6  P.M. 

—  7  Beginning  at  the  scale-mark  —6,  and  counting 


14  ALGEBRA. 

11  degree-spaces  upwards,  we  reach  the  scale-mark  +5; 
and  counting  from  the  latter  9  degree-spaces  downwards, 
we  reach  the  scale-mark  —  4.  Hence,  the  temperature  at 
noon  is  -f  5°,  and  at  6  p.m.  —  4°. 

EXERCISES. 

46.  1.  At  7  A.M.  the  temperature  is  —  8°  ;  at  noon  it  is 
7°  warmer,  and  at  6  p.m.  it  is  3°  colder  than  at  noon. 
Required  the  temperatures  at  noon  and  at  6  p.m. 

2.  A  certain  city  was  founded  in  the  year  151  B.C.,  and 
was  destroyed  203  years  later.  In  what  year  was  it  de- 
stroyed ? 

3.  At  7  A.M.  the  temperature  is  +  4° ;  at  noon  it  is  lO'' 
colder,  and  at  6  p.m.  it  is  6°  warmer  than  at  noon.  Re- 
quired the  temperatures  at  noon  and  at  6  p.m. 

4.  What  is  the  difference  in  latitude  between  two  places 
whose  latitudes  are  -|-56°  and  —31°? 

5.  A  man  has  bills  receivable  to  the  amount  of  $2000, 
and  bills  payable  to  the  amount  of  $3000.  How  much  is  he 
worth  ? 

6.  At  7  a.m.  the  temperature  is  —  3°,  and  at  noon  it  is 
-f  11°.     How  many  degrees  warmer  is  it  at  noon  than  at  7 

A.M.? 

7.  What  is  the  difference  in  longitude  between  two  places 
whose  longitudes  are  +  25°  and  —  90°  ? 

8.  The  temperature  at  6  a.m.  is  —  7°,  and  during  the 
morning  it  grows  warmer  at  the  rate  of  3°  an  hour.  Re- 
quired the  temperatures  at  8  a.m.,  at  9  a.m.,  and  at  noon. 

47.  The  absolute  value  of  a  quantity  is  the  number  repre- 
sented by  the  quantity,  taken  independently  of  the  sign 
affecting  it. 

Thus,  the  absolute  value  of  —  5  is  5. 


ADDITION.  15 


.   II.  ADDITION. 

48.  Addition,  in  Algebra,  is  the  process  of  collecting  two 
or  more  quantities  into  one  equivalent  expression,  called  the 
sum. 

Thus,  the  sum  of  a  and  b  is  a  +  b  (Art.  8). 

49.  If  either  quantit}^  is  negative,  or  a  polynomial,  it 
should  be  enclosed  in  a  parenthesis  (Art.  20)  ;  thus. 

The  sum  of  a  and  —  6  is  indicated  by  a  -|-  (—  6). 
The  sum  of  a  —  b  and  c  —  d  is  indicated  by 

(a  —  6)  4-  (c  —  d) . 

50.  Required  the  sum  of  a  and  —  b. 

Using  the  interpretation  of  negative  quantities  as  ex- 
plained in  Art.  44,  if  a  man  incurs  a  debt  of  $100,  we  may 
regard  the  transaction  either  as  adding  —$100  to  his  prop- 
erty, or  as  subtracting  $100  from  it      That  is. 

Adding  a  negative  quantity  is  equivalent  to  subtracting  a 
positive  quantity  of  the  same  absolute  value  (Art.  47). 

Thus,  the  sum  of  a  and  —  &  is  obtained  by  subtracting  b 

from  a ;  or, 

a  +  (  —  Z>)  =  a  —  6. 

51.  It  follows  from  Arts.  48  and  50  that  the  addition  of 
monomials  is  effected  by  uniting  the  quantities  with  their 
respective  signs. 

Thus,  the  sum  of  a,  —  6,  c,  —  d,  and  —  e,  is 

a  —  b-]-c  —  d  —  e. 

It  is  immaterial  in  what  order  the  terms  are  united,  pro- 
vided each  has  its  proper  sign.     Thus,  the  above  result  may 

also  be  expressed  ^ 

c  -{-  a—e  —  d  —  b, 

—  d  —  6  +  c  —  e+a,  etc. 


16  ALGEBHA. 

ADDITION  OF  SIMILAR  TERMS. 

52.  1.  Required  the  sum  of  5  a  and  Sa. 

6  a  signifies  a  taken  5  times  (Art.  12),  and  3  a  signifies  q 
taken  3  times.  We  have,  therefore,  a  taken  in  all  8  times, 
or  8  a.     That  is, 

5a  +  3a  =  8a. 

2.    Required  the  sum  of  —  5  a  and  —  3  a. 

—  5  a  signifies  a  taken  5  times  subtractively  (Art.  25) ,  and 
—  3a  signifies  a  taken  3  times  subtractively.  We  have, 
therefore,  a  taken  in  all  8   times  subtractively,   or   —8  a. 

That  is, 

—  5a  —  3a  =  —  8a. 

Therefore, 

To  add  two  similar  (Art.  31)  terms  of  like  sign,  add  the 
coefficients,  affix  to  the  result  the  common  symbols,  and  pi'ejix 
the  common  sign. 

53.  1.  Required  the  sum  of  8  a  and  —  5  a. 

Since  8  a  is  the  sum  of  3  a  and  5  a  (Art.  52,  1),  the  sum 
of  8a  and  —5a  is  equal  to  the  sum  of  3a,  5a,  and  —5a, 
which  is 

3a  +  5a  — 5a.  (Art.  51.) 

But,  by  Art.  26,  5a  and  —5a  cancel  each  other,  leaving 
the  result  3  a. 

Hence,  8a+(— 5a)  =  3a. 

2.  Required  the  sum  of  —Sa  and  5a. 

Since  —  8a  is  the  sum  of  —  3a  and  —  5a  (Art.  52,  2) ,  the 
sum  of  —8a  and  5a  is  equal  to  the  sum  of  —3a,  —5a, 
and  5  a,  which  is 

—  3a  — 5a4-5a,  or  —3a. 

Hence,  (  — 8a)  +  5a  =  —  3a. 


ADDITION.  17 

Therefore, 

To  add  two  similar  terms  of  unlike  sign,  subtract  the  less 
coefficient  from  the  greater,  affix  to  the  result  the  common  sym- 
bols, and  prefix  the  sign  of  the  greater  coeffixiient. 

Note.  A  clear  understanding  of  the  nature  of  the  processes  in  Arts. 
52  and  53  may  be  obtained  by  comparing  them  with  the  following,  the 
negative  quantities  being  interpreted  as  explained  in  Art.  44. 

1.  If  a  man  owes  $5,  and  incurs  a  debt  of  §3,  he  will  be  in  debt  to 
the  amount  of  $8.     That  is,  the  sum  of  —  $ 5  and  - $3  is  —  6 8. 

2.  If  a  man's  assets  amount  to  $8,  and  his  liabilities  to  $6,  he  is 
worth  $3.     That  is,  the  sum  of  §8  and  -  $5  is  $3. 

3.  If  a  man's  liabilities  amount  to  $8,  and  his  assets  to  $5,  he  is  in 
debt  to  the  amount  of  $3.     That  is,  the  sum  of  —  $8  and  $ 5  is  —.$3. 

EXAMPLES. 
54.    Add  the  following  : 

1.  11  and  -5. 

2.  -  13  and  3. 

3.  12  and  -1. 

4.  —  4  and  —  7. 

5.  —2a  and  la. 

6.  b  and  -36. 

13.  Required  the  sum  of  2  a, 
Since  the  order  of  the  terms  is  immaterial  (Art.  51),  yre 

may  add  the  positive  terms  first,  and  then  the  negative,  and 
finally  combine  these  results  by  the  rule  of  Art.  53. 

The  sum  of  2  a,  3  a,  and  6  a  is  11a. 

The  sum  of  —a  and  —  12a  is  —  l3a. 
Hence,  the  required  sum  is  11a  —  13  a,  or  —  2a.  Ans. 

Add  the  following : 

14.  7a,  —  a,  and  —3a.       15.    —  6m,  m,  —  11m,  and  5m. 

16.    13  a6,  —  7a6,  —  8a6,  and  —%ab. 


7. 

—  11m  and  —Sm. 

8. 

be  and  166c. 

9. 

—  2  ax  and  7  ax. 

10. 

-Sa'b^  and— a'b\ 

11. 

1 2  mn2  and  -19mn\ 

12. 

—  13  abc  and  22  a6c. 

—  a, 

,3a,—  12a,  and  6a. 

18  ALGEBRA. 

17.  ln\  -n\  -3ii2,  \\n\  and  -\0n\ 

18.  13  ax^,  —  aa^,  —  20aa^,  6aa^,  and  —5aaf. 

If  the  terms  are  not  all  similar,  we  may  combine  the  simi- 
lar terms,  and  unite  the  others  with  their  respective  signs. 

19.  Required  the  sum  of  12a,  —5x,  —Sy,  —5a,  8a:, 
and  —3x. 

The  sum  of  12  a  and  —  5a  is  7a. 
The  sum  of  —  5  a:,  8  x,  and  —  3  a?  is  0. 

Hence,  the  required  sum  is  7a  — 3y.  Ans. 
Add  the  following : 

20.  6 ax,  —115,  —ax,  and  65. 

21.  2a,  55,  -3c,  -85,  and  9c. 

22.  5  m,  —  2n^,  n,  —2  m,  —5^,  and  37i^ 

23.  3a;,  —y,  ^x,  6,  —8y,  —2x,  4^,  and  —5. 

ADDITION  OF  POLYNOMIALS. 

55.  A  polynomial  may  be  regarded  as  the  sum  of  its 
monomial  terms  (Art.  51).  Thus,  2a  — 35  +  4c  is  the  sum 
of  the  terms  2  a,  —3  5,  and  4  c. 

Hence,  the  addition  of  two  or  more,  polynomials  is  effected 
by  uniting  their  terms  with  their  respective  signs. 
Thus,  the  sum  of  a  —  5  and  c  —  d  \^  a  —  h  -\-  c  —  d. 

56.  Required  the  sum  of  6a  — 7a;,  3a;  — 2a4-3y,  and 
2  a;  —  a  —  mn. 

It  is  convenient  in  practice  to  set  the  expressions  down 
one  underneath  the  other,  similar  terms  being  in  the  same 
vertical  column.     Thus, 

6a—  7a; 

—  2a  +  3a;  +  3i/ 

—  a-\-2x  —mn 

3a  — 2x  + 3?/  — mn,  Ans. 


ADDITION.  19 

From  the  above  principles  we  derive  the  following  rule : 

To  add  tico  or  more  expressions,  set  them  down  one  under- 
neath the  other,  similar  terms  being  in  the  same  vertical  colunni. 
Find  the  sum  of  the  terms  in  each  column,  and  unite  the  results 
with  their  respective  signs. 

EXAMPLES. 
57.  Add  the  following : 


1. 

2. 

3. 

2a-lx 

-Sab-\-2cd 

—  11a—    5mp^ 

—  a  +  4tx 

—  7ab-{-Scd 

8a-{-nmp^ 

a+    X 

4a6-6cd 

—    da—    7  mp^ 

4.  2a  — 36  4- 5c  and  6  — 5c  +  2d. 

6.  9  mn^  +  x^y,  —  mn^  -f-  3  x^y,  and  —  6  mn^  —  7  x^y. 

6.  aP-2ab-\-b\  a- -{- 2  ab -\- b\  and2a2-262. 

7.  3a2  +  2a6H-462,  oa^-Sab-\-b\  and  -6a'-^Dab-5b\ 

8.  6a^-7x-4:,x^-x-2,SindSx-dxr-x^. 

9.  4  mn  -f  3  a6  —  4  c,  3  a;  —  4  a6  +  2  mn,  and  3  m^  —  4  a:. 

10.  3x  —  2y  —  z,  Gy  —  5x  —  Iz,  82  —y  —  x,  and  4a;— 92/. 

11.  Qx  —  ^y-\-lm,  2n  —  x  +  y,  2y  —  4:X  —  5m  —  9n, 

and  m  —  2  x. 

12.  2if^-6x^-x-\-7,  3x^  —  2-6ar  +  8x,  a;  +  3x'-4, 

and  1  +  2  a,'^  —  5  x. 

13.  2a-36  +  4d,     26-3d  +  4c,     2<i  -  3c +  4a +  46, 

and  2  c  —  3  a. 

14.  2a^-a^b-2b\    Sa"" -8ab^  -  3b\    Sa'b  -ab^+b\ 

and6a62_2a2&-5a3. 

16.    4a.'3-10a3-5aa;2  +  6a2a;,  Ga^  +  30^  +  4aar'  + 2a2a;, 

-17a.-3+19aa;2_i5^2^^and6a.'3+7a2a;+5a«-18aa;2^ 


20  ALGEBRA. 

III.   SUBTRACTION. 

58.  Subtraction,  in  Algebra,  is  the  process  of  taking  one 
quantity  from  another. 

The  Subtrahend  is  the  quantity  to  be  subtracted. 
The  Minuend  is  the  quantity  from  which  it  is  to  be  sub- 
tracted. 

The  Remainder  is  the  result  of  the  operation. 

59.  It  is  evident  from  the  above  that  the  minuend  is 
equal  to  the  sum  of  the  subtrahend  and  the  remainder. 

60.  Let  it  be  required  to  subtract  —  h  from  a. 

Using  the  interpretation  of  negative  quantities  as  explained 
in  Art.  44,  if  a  man  cancels  a  debt  of  $100,  we  may  regard 
the  transaction  either  as  subtracting  —  $100  from  his  prop- 
erty, or  as  adding  $100  to  it.     That  is. 

Subtracting  a  negative  quantity  is  equivalent  to  adding  a 
positive  quantity  of  the  saine  absolute  value. 

Thus,  to  subtract  —  b  from  a,  we  add  5  to  a  ;  or 

a  — (—6)  =  a  4-  6. 

Hence,  to  subtract  one  quantity  from  another^  change  the 
sign  of  the  subtrahend.,  and  add  the  result  to  the  minuend. 

61.  1.   Subtract  5  a  from  2  a. 

By  Art.  60,  the  result  is  equal  to  the  sum  of  —  5a  and  2a, 
which  is  —3  a. 

2.  From  —  2a  subtract  5a. 

The  result  is  equal  to  the  sum  of  —  2  a  and  —  5  a,  or  —  7  a. 

3.  From  5  a  take  —  2  a. 

Result,  5a4-2a,  or7a. 

4.  From  —2a  take  —5a. 

Result,  —  2  a  -f  5  a,  or  3  a. 


S13BTKACTI0N.  21 


EXAMPLES, 
62.    Subtract  the  following  : 


1.    -3  from  11. 

3. 

—  8  from  —  3.         5.    23  from  10. 

2.   16  from  —5. 

4. 

-  11  from  -  17.     6.    -  13  from  11, 

7.                   8. 

9.                    10.                    11. 

27a                   17aj 

—  13y             —  lOmn               oa^b 

ir>a               —11a; 

Ay              -ISmn              Ua-b 

12.  From  9  ab  take  -  2  ab.    16.  From  -  ar^/  take  5x^y\    ' 

13.  From  xy  take  —  cd.         17.  From  —  70 abc  take  —  b2abc. 

14.  Froml7mHake41??i^    18.  From  -  7 ?w- take  -  8 n^. 

15.  From -5a;  take  3.  19.  From -33a;»2/^  take  19ar^2/*. 

20.  From  bab  take  the  sum  of  9a6  and  —2ab. 

21.  From   the    sum   of    —  Ua:^   and   Sa?   take  the  sum  of 

-  10  ar'  and  4ar'. 

SUBTRACTION    OF    POLYNOMIALS. 

63.  When  the  subtrahend  is  a  polynomial,  each  of  its 
terms  is  to  be  subtracted  from  the  minuend.     Hence, 

To  subtract  one  polynomial  from  another^  change  the  sign  of 
each  term  of  the  subtrahend,  and  add  the  result  to  the  minuend. 

It  will  be  found  convenient  to  place  the  subtrahend  under 
the  minuend,  similar  terms  being  in  the  same  vertical  column. 

64.  1.   Subtract  ba?y  —  3ab  -{-  m^  from  3  x^y  —  2  a6  -f-  4  ?i. 

Changing  the  sign  of  each  term  of  the  subtrahend,  and 
adding  the  result  to  the  minuend,  we  have 
3ar>-2a6  +  4?i 
—  ba?y-\-3ab  —  m^ 


—2a^y-\-    ab-\-4:n  —  m^ 


22  ALGEBRA. 

Note.  The  student  should  endeavor  to  perform  mentally  the  opera- 
tion of  changing  the  sign  of  each  term  in  the  subtrahend,  as  shown  in 
the  following  example : 

2.    From  5a^-7b^~2a^b  subtract  Sa'b^  4:ab'-2b'-^aK 

6a^-2a'b  -^7b^ 

a^-\-3a^b-Aab'-2b^ 

EXAMPLES. 
Subtract  the  following : 

3.  4. 

ab  -\-    cd—    ax  7x-\-by  —  Sa 

Aab  —  Scd  —  4:ax  x  —  7y-i-6a  —  4: 

5.  From  a  —  6  +  c  take  a  +  b  —  c. 

6.  From  a^-^2ab-\-b^  take  a'-2ab-{-  b\ 

7.  From  7abc—nx+oy—4:8  take  lla6c4-3a;-f-72/+100. 

8.  Subtract  3m-\-y-—5a  —  7  from  5m  — 3/+  7a  —  6. 

9.  Subtract  17a^ +  5^/^- 4a5 -f  7  from  31a^-3/-f  a6. 

10.  From  6a  +  36— 5c-fl  subtract  6a— 36  —  5c. 

11 .  From  Sm  —  5n  +  r  —  2s  take  2 /•  +  3 n  —  m  —  5 «. 

12.  Take  4a  —  5  +  2c  —  5c?  from  d  —  Sb  +  a  —  c. 

13.  From  m^  +  3  w^  subtract  —  4  m^  —  6  w^  +  71  a;. 

14.  From46  — 3  6  — 5d  +  2ic  take  3a -\- 8d—b  —  6c. 

16.    From  a  —  b  —  c  take  the  sum  of  — 2a  +  6+c 
and  a^b  -^c. 

16.  Froma;*  +  2a^-3a;  +  4  take  3a^  + 30^+ 5a;- 7. 

17.  From  4  a^  -  3  afe^  _  5  6^  subtract  6  a^b  -  aft^  +  4  b\ 

18.  Froma^  — 8H-2a*-3aHake  6a- U  —  Sa^  — 2aS 


SUBTRACTION.  23 

19.    Take  2x^  —  y'^  from  the  sum  of  x^  —  2xy  -^^y^ 
and  xy  —  i  y^. 

—       20.    From  the  sum  of  a;-}- 22/  — 32;  and  3y —  4ic  + a 
take  z  —  bx-\-by. 

21.  From  7a3  +  3-5(X^  +  a-5a2 

subtract  2a—^o?—2a?-\-^  —  \la\ 

22.  From —72/^  +  3 a^2/  —  2it'*  + 6 a.-^/^ 

subtract  Sa^y  —  2  xy^  -{-  a:^  —  9  y^. 

23.  From  the  sum  of  2a:^  —  x-\-  5  and  ar^  +  8 a;  —  11  take  the 

sum  of  ay^  —  9x^  —llx  and  —  42/-^  +  3 a.*^  —  6. 

-^     24.    From  the  sum  of  a-  -f-  a6  +  6^  and  a^  —  4  a6  -f  5  6^  take 
the  sum  of  4a2  + 76"- 2a6  and  Sab-a--2b^. 

"i     26.   From3ar^-7?/-2+a:2/-5/ 

subtract  —  5xy  -\-(jx  —  2x^  —  S-\-  2y^. 

26.  From  3ar'  — 8a.'*  +  3a.'^  — 5a^  — 2aj 

subtract  -  3a^  +  4a^+ 6a^- 6a;  + 2. 

27.  From  the  sum  of  2 3i?  —  3?y  —  6 xif  and  3a^y  —  5xy^  —  4y^ 

take  the  sum  of  —2a^—7ofy—6y^  and  —  6a:y^+5?/*. 

-^      28.    From  the  sum  of  a*  —  1  and  2a^  —  lOa^— 7a  subtract  the 
sum  of  —3a*-\-2a^  —  5a  and  — 5 a^— 12 a^ 4-3. 

Note.  In  Arithmetic,  addition  always  implies  augmentation,  and 
subtraction  diminution.  In  Algebra  this  is  not  always  the  case;  for 
example,  in  adding  —  2  to  5,  the  sum  is  3,  which  is  less  than  5.  Again, 
in  subtracting  —  2  from  5,  the  remainder  is  7,  which  is  greater  than  5. 

Thus  the  terms  Addition,  Subtraction,  Sum,  and  Remainder  have  a 
much  more  general  signification  in  Algebra  than  in  Arithmetic. 


,24  ALGEBRA. 


IV.   USE    OF    PARENTHESES. 

65.  The  use  of  parentheses  (Art.  20)  is  very  frequent  in 
Algebra,  and  it  is  necessary  to  have  rules  for  their  removal 
or  introduction. 

66.  The  expression 

2a-3b+{bb-G-\-2d) 

indicates  that  the  quantity  6b  —  G-\-2d  is  to  be  added  to 
2a  — Sb.     If  the  addition  be  performed,  we  obtain  (Art.  55) 

2a-3b-\-5b-G  +  2d. 

Again,  the  expression 

2a_35-(56-c  +  2d) 

indicates  that  the  quantity  5b  —  c-\-2d  is  to  be  subtracted 
from  2  a  — 3  b.  If  the  subtraction  be  performed,  we  obtain 
(Art.  63)  2a-36-56  +  c-2d. 

67.  It  will  be  observed  that  in  the  first  case  the  signs  of 
the  terms  within  the  parenthesis  are  unchayiged  when  the 
parenthesis  is  removed ;  while  in  the  second  case  the  sign 
of  each  term  within  is  changed,  from  +  to  — ,  or  from  — 
to  +. 

We  have  then  the  following  rule  for  removing  a  paren- 
thesis : 

A  parenthesis  preceded  by  a  -\-  sign  may  be  removed  without 
altering  the  signs  of  the  enclosed  terms. 

A  parenthesis  preceded  by  a  —  sign  may  be  removed,  if  the 
sign  of  each  enclosed  term  be  changed,  from  ■\-  to  —,  or  from 
—  to  +. 

68.  Since  the  brackets,  the  braces,  and  the  vinculum 
(Art.  20)  have  the  same  signification  as  the  parenthesis, 
the  rule  for  their  removal  is  the  same. 


PARENTHESES.  25 

It  should  be  observed  in  the  case  of  the  vinculum  that  the 
sign  apparently  prefixed  to  the  first  term  underneath,  is  in 
reality  the  sign  of  the  vinculum.  Thus,  -j-  a  —  6  and  —a  —  b 
are  equivalent  to  +  (a  —  6)  and  —  (a  —  6) ,  respectively. 

EXAMPLES. 
69.    1.  Remove  the  parentheses  from 

2a-36-(5a-46)-f(4a-6). 

By  the  rule  of  Art.  67,  the  expression  becomes 

2a  —  36  —  5a  +  46-|-4«  —  6  =  a,  Ans. 

Parentheses  are  often  found  enclosing  others.  In  this 
case  they  may  be  removed  in  succession  by  the  rule  of  Art. 
67,  and  it  is  better  to  remove  first  the  innermost  pair. 

2.  Simplify  the  expression 


We  remove  the  vinculum  first,  and  the  others  in  succession. 


Thus, 


4a;—  53a;  +  (— 2if  — a;  — a)5 
=  4.x-\Zx-\-{-2x-x-^a)\ 
=  A:X  —  \^x  —  2  X  —  X  +  a\ 
=  4a;  — 3a:  +  2fl;-f-.T  — a  =  4a;  — a,  Ans. 

Reduce  the  following  expressions  to  their  simplest  forms 
by  removing  the  parentheses,  etc.,  and  uniting  similar  terms  : 

3.  a-(&-c)  +  (-cZ  +  e). 

4.  5a;-^2a;-32/^-[-2a;  +  4?/]. 

5.  a-~h-{-c  —  a-\-h  —  c  —  c  —  h  — a. 


m^. 


6.  m-  —  2 ?i  +  J rt  —  ?i  +  3 m^J  —  5a  +  3?i 

7.  ar-V-  {or  -  2ab  +  5")  -  [a-  +  2a6  +  6-]. 

8.  3a-(2a- Ja  +  2;). 


26 


) 

ALGEBRA. 

9. 

a- 

~(b  +  \ 

-e  +  d}- 

-e). 

10. 

a  - 

-[(-b 

+  c)-(d- 

-«)]• 

11.  Sx-l2y-{-x-y']-\-[3y-2x-\~yl^. 

12.  Ux-{ox-9)-\4.-Sx-(2x-3) 

13.  2m-[n-j3m-(2n-m)J]. 


14.  Sx  — (5 X -{-[_  — 4:X  —  y  —  x'])  —  (—x  —  3y). 


16.  3c-f-(2a-[5c-j3a  +  c-4a5]), 


16.  5a-(4a-J-3a-[2a-a-l]J). 

17.  8x-l5x-(Sx-A)-\7x-i-{-9x-{-2)l'] 


i  18.  2m  -  [3m  -  Jm  -  (2m  -  3m  +  4)  ;  -  (5m  ~2)]. 
19.  c  -  [2  c  -  (6  a  -  6)  -  J  c  -  (5  a  +  2  &)  -  (a  -  3  6)  J  ] . 


/     20.  Sa-\b-lb-(a-j-b)-\-b-{b-a-b)l']\. 

70.  To  enclose  any  number  of  terms  in  a  parenthesis,  we 
take  the  converse  of  the  rule  of  Art.  67  : 

Any  number  of  terms  may  be  enclosed  in  a  parenthesis  pre- 
ceded by  a  -\-  sign,  without  altering  their  signs. 

Any  number  of  terms  may  be  enclosed  in  a  parenthesis  pre- 
ceded by  a  —  sign,  if  the  sign  of  each  term  be  changed,  from 
-{•  to  —,  or  from  —  to  -\-. 

71.  1 .  Enclose  the  last  three  terras  of  a  —  b  +  c  —  d-\-e 
in  a  parenthesis  preceded  by  a  —  sign. 

Result,  a  —  b—  (—  c  -j-  d  —  e) , 

In  each  of  the  following  expressions,  enclose  the  last 
three  terms  in  a  parenthesis  preceded  by  a  —  sign  : 

2.  a-\-b-{-c-{-d.  5.   oc^y  —  nc^y'^  —  x^  +  y^. 

3.  3a-2&  +  5c-4d.  6.    x' -Zo? -\-2x' -bx-8. 

4.  m^  +  5m2-6m  +  3.  7.    a^ -b^  -  c^ -{-2a'b -\-2ac. 

8.  In  each  of  the  above  results,  enclose  the  last  two  terms 
in  an  inner  parenthesis  preceded  by  a  —  sign. 


MULTIPLICATION.  27 


V.   MULTIPLICATION. 

72.  Multiplication,  in  Algebra,  is  the  process  of  taking 
one  quantity  as  many  times  as  there  are  units  in  another. 

Thus,  the  multiplication  of  a  b}'  6,  which  is  expressed  ab 
(Art.  10),  signifies  that  the  quantity  a  is  to  be  taken  b 
times. 

73.  The  Multiplicand  is  the  quantity  to  be  multiplied  or 
taken. 

The  Multiplier  is  the  quantity  which  shows  how  many 
times  it  is  to  be  taken. 

The  Product  is  the  result  of  the  operation. 

The  multiplicand  and  multiplier  are  called /acfors. 

74.  In  Arithmetic,  the  product  of  two  numbers  is  the 
same  in  whatever  order  they  are  taken  ;  thus,  we  have  3x4 
or  4  X  3,  each  equal  to  12. 

Similarly,  in  Algebra,  we  have  a  x  6  or  6  x  a,  each  equal 
to  ab. 

That  is,  the  product  of  the  factors  is  the  same  in  whatever 
order  they  are  taken. 

75.  Required  the  product  of  c  and  a  —  b. 

In  Arithmetic,  if  we  wish  to  multiply  87  by  98,  we  may 
express  the  multiplier  in  the  form  100  —  2  ;  we  should  then 
multiply  87  by  100,  and  afterwards  by  2,  and  subtract  the 
second  result  from  the  first. 

Similarly,  in  Algebra,  to  multiply  c  by  «  —  5,  we  should 
multiply  c  by  a,  and  afterwards  by  6,  and  subtract  the  sec- 
ond result  from  the  first.     Thus,  the  required  product  is 

ac  —  be. 

76.  Required  the  product  of  a  —  6  and  c  —  d. 

As  in  Art.  75,  we  should  first  multiply  a  —  b  by  c,  and 


28  •  ALGEBRA. 

afterwards  by  d,  and  subtract  the  second  result  from  the 
first. 

The  product  of  a  —  &  and  c  is  ac  —  he  (Art.  75) . 

The  product  of  a  —  h  and  d  is  ad  —  hd. 

Subtracting  the  second  result  from  the  first,  the  required 
product  is  „<.-6c-«d  +  M. 

77.  We  observe,  in  the  preceding  article,  that  the  product 
is  formed  by  multiplying  each  term  of  the  multiplicand  by 
each  term  of  the  multiplier,  with  the  following  results  in 
regard  to  signs : 

The  product  of  the  terms  +  a  and  +  c  gives  the  term  -f-  ac. 
The  product  of  the  terms  —  h  and  +  c  gives  the  term  —  he. 
The  product  of  the  terms  +  a  and  —  d  gives  the  term  —  ad. 
The  product  of  the  terms  —  h  and  —  d  gives  the  term  +  hd. 

From  these  considerations  we  may  state  what  is  called  the 
Rule  of  Signs  in  Multiplication,  as  follows  : 

+  multiplied  hy  +  •>  c^'^^d  —  multiplied  hy  — ,  produee  +  ; 
+  multiplied  hy  — ,  arid  —  multiplied  by  -f ,  produce  — . 

Or,  as  it  is  usually  expressed  with  regard  to  the  product  of 
two  terms. 

Like  signs  produce  + ,  and  unlike  signs  produce  — . 

78.  Required  the  product  of  7a  and  2h. 

Since  the  factors  may  be  written  in  any  order  (Art.  74) , 

we  have 

7 a  X  2h  =  7  X  2  X  a  X  h  =  lAah. 

That  is,  the  coefficient  of  the  product  is  the  product  of  the 
coefficients  of  the  factors. 

79.  Required  the  product  of  a^  and  a^. 

By  Art.  13,  a^  =  a  x  a  x  a,  and  a^==axa.     Hence, 
a^xa^  =  axaxaxaxa  =  a^. 


MULTIPLICATIOK.  29 

That  is,  the  exponent  of  a  letter  in  the  product  is  the  sum  of 
its  exponents  in  the  factors. 

Thus,  a'xa^Xa  =  a'+^^  =  aK 

MULTIPLICATION  OF  MONOMIALS. 

80.  We  derive  from  Arts.  77,  78,  and  79  the  following 
rule  for  the  product  of  two  monomials  : 

To  the  product  of  the  coefficients  annex  the  literal  quantities^ 
giving  to  each  letter  an  exponent  equal  to  the  sum  of  its  expo- 
nents in  the  factors.  Make  the  product  -\-  when  the  factors 
have  the  same  sign,  and  —  when  they  have  different  signs. 

EXAMPLES. 

1.  Multiply  2  a'' by  7a\ 

By  the  rule,  2a^  x  7a*  =  14a^+^''  =  14a»,  Ans. 

2.  Multiply  a''^6-c  by  -5a'bd. 

a^b^c  X  (  —  5  a-bd)  =  —  5  a^b^cd,  Ans. 

3.  Multiply  -7. T"*  by  5 ar\ 

-  7  a;'"  X  6x^  =  -  35a;"'+^  Ans. 

4.  Multiply  —11a;'"  by  —Sx'". 

-nx'^x{-Sx^)  =  88x-'^,  Ans. 

Multiply  the  following : 

6.  13  by  -19.  11.  -ll^i^yby  -57i<^. 

6.  -  18  by  12.  12.  -  6a^bc  by  a^bm. 

7.  -22  by  -51.  13.  -  12a2a;  by  -20;^. 

8.  15m«n«  by  3mn.  14.  -  2a'"6"  by  5a^6". 

9.  17a6c  by  —8 a5c.  15.  3 aV/ by  1 1  aa^y . 
10.  -17aVby3aV.  16.  3a'"6"  by  -5a"5^ 


80  ALGEBRA. 

It  is  evident  from  the  Rule  of  Signs  (Art.  77)  that  the 
product  of  three  negative  terms  is  negative  ;  of  four  negative 
terms,  positive  ;  and  so  on. 

Hence  the  product  of  three  or  more  monomials  will  be  pos- 
itive or  negative,  according  as  the  number  of  negative  factors 
is  odd  or  even. 

17.    Required  the  product  of  —  20%^  Qthd^^  and  —  Ic^d. 

-  2  a^d^  X  6  6c^  X  -  7  c^d  =  84  a^ftVd,  Ans. 

In  this  case  the  product  is  positive,  as  there  are  two  nega- 
tive factors. 

Multiply  the  following : 

18.  5a,  —66,  and  7c. 

19.  -2a2,  -Ua^,  and  -9  a. 

20.  -  3  aV,  -  2  6r ,  and  7  c^. 

21.  ^x'^y^,  —  ic"2/V,  and  \hyh\ 

22.  -2a,  -3a2,  -4a^  and  -5a^ 

23.  -  a^hc,  2  V^cd,  —  5  d'cd,  and  -  3  ahH\ 

24.  —  7m'*a^,  m^aj^,  2a^,  and  —%my^\ 

25.  6072/^,  —o?z^  32/V,  —  2fl7r^,  and  —^yz, 

MULTIPLICATION  OF  POLYNOMIALS    BY  MONOMIALS. 

81.  In  Art.  75  we  showed  that  the  product  of  a  —  6  and  c 
was  ac  —  bc.  We  have  then  the  following  rule  for  the  prod- 
uct of  a  polynomial  by  a  monomial : 

Multiply  each  term  of  the  multiplicand  by  the  multiplie'i\ 
and  connect  the  results  with  their  proper  signs. 

EXAMPLES. 
1.  Multiply  2ic2 _  5a; _  7  by  8aj3. 
By  the  rule,  the  product  is  1 6  a^  —  40a/'''  —  560.-^,  Ans, 


MULTIPLICATION.  ^         31 


2.   Multiply  -  5  ab^  by  3  a^ft  -  4  aW. 
3a26_4a63 


Multiply  the  following : 

3.  3a;  — 5  by  4a;.  8.    m^ +  mn-\-n^hy  m^n^. 

4.  a^h  +  a&2  by  -  ah.  9.    -  2m  by  Sm^-5mn-n\ 

5.  Sa^bc -d  by  5a(P.  10.    -  a;.^- lOa^^+S  by  -  2a;3^ 

6.  a.-2-2a;-3  by  -4ar.       11.    a^ +  l^ab  -  6b^  by  4:ab\ 

7.  -2a^by3a^-^ex-7.   12.    -Ga^c  by  5  -  6ac-8a^ 

13.  5a;3_4a^_33,_^2  by  -Gar'. 

14.  a^b^  by  a^  -  3  a^b  +  3  ab^  -  6^. 

MULTIPLICATION  OF  POLYNOMIALS  BY  POLYNOMIALS. 

82.  In  Art.  7G  it  was  shown  that  the  product  of  a  —  b  and 
c  —  d  was  oo  —  6c  —  afZ  -f  bd.  We  have  then  the  following 
rule  for  the  product  of  two  polynomials : 

Multiply  each  term  of  tJie  multiplicand  by  each  term  of  the 
multiplier^  and  add  the  partial  products. 

EXAMPLES. 

1.    Multiply  3a  — 26  by  2a  — 56. 

In  accordance  with  the  rule,  we  multiply  3  a  — 26  by  2a, 
and  then  by  —56,  and  add  the  partial  products.  A  convenient 
aiTangement  of  the  work  is  shown  below,  similar  terms  being 
in  the  same  vertical  column. 

3a  —    26 
2a  -   56 


Qa^-~   Aab 

-15a6  +  lQ6'^ 
6a2-19a6  +  1062,  Ans. 


32  ALGEBRA.  I 

2.  Multiply  x^  +  l  —  x^  —  xhyx-\-l. 

It  is  convenient  to  have  both  multiplicand  and  multiplier 
arranged  in  the  same  order  of  powers  (Art.  36) ,  and  to  write 
the  partial  products  in  the  same  order. 

Arranging  the  expressions  according  to  the  ascending 
powers  of  x,  we  have 

l—x  +  x^  —  a^ 

l-\-x 

1  —x-i-x^  —  a^ 

x  —  x^-\-x^  —  x* 
1  —x'^,  Ans. 

3.  Multiply  6  a6- 8  52  +  4  a^  by  -4b' -^2a^ -Sab. 

Arranging  according  to  the  descending  powers  of  a,  we 

have 

4a2  +  6a&-862 

20" -Sab -W 

Sa^  +  12a%-16a'b' 

-12a^b-  18a^b'-{-24:ab^ 

-Ua^b'-24:ab^-{-S2b* 

8a*  -60aW  +32&S  Ans. 

Note.  The  correctness  of  the  answers  may  be  tested  by  working  the 
examples  with  the  muhiplicand  and  multipHer  interchanged. 

Multiply  the  following : 

4.  3a; +  2  and  5a; -7.  6.    3a-26  and -2a  +  4&. 

5.  6a;— 5  and  3  ~2«.  7.    3  —  Sa;^/ and  —  6  — lOa^y. 

8.  a^-\-ab-{-  b'  and  b  —  a. 

9.  2a^b-S  ab^  and  5a'b  +  6  ab\ 

10.  1  +  a;  4-  a^  +  a^  and  ax  —  a. 

11.  3a;2-2a;2/-/ and  2a;-42/. 

12.  m^  —  mn  —  Sn^  and  2  m^  —  6  mn. 

13.  a^-h2a;+l  anda;2-2a;-h3. 


MULTIPLICATION.  33 

14.  ba'  +  ib'-Sab  and  6a- 5b. 

15.  4a;^4-6ic-7  and  2x2-3. 

16.  a-hb  —c  and  a  —  b-^c. 

17.  2x^-3x-{-5  sindx^-j-x-1. 

18.  3x'-7x-\-4:B.nd2x'-i-9x~5. 

19.  2x3-3x2-5x-l  and3a;-5. 

20.  6  m  —  2  7n,2  _  5  _  ^^^3  j^j^^i  ^^^2  _f_  2  0  —  2  m. 

21.  2a;3  4-5x2_g^_7  ^^j^^^  ^_  .^_^^^ 

22.  a'b  -  a'b'  -  4  ab^  and  2  a^ft  -  ab^ 

23.  a;'"+V  __  3 ^y--^  and  4x'»+y  -  4xy. 

24.  a;2  -^.y^  —  xy  and  xy  +  /  -}-  ^2^ 

25.  2a6  +  62_^4a2and4a2_2a6  +  62. 

26.  6.T*-3a^-x2+6x-2and2.x'2  +  a;  +  2. 

27.  m*  -  ^i^/i  +  7n-7r  -  wiji^  _^  ^^4  ^^^^  ^^^2  _  2  ^^^^^  _  3  ^2^ 

28.  27x^H-9x22/  +  3V  +  2/-^and9a.-2_6x?/4-2/^ 

29.  a^-Sa'b-\-3ab^-b^andd'-2ab-^bK 

30.  a;2  +  i/2  +  2;2^a^_2^2_;2a;anda;  +  ?/-|-«. 

31.  2a:3-3x'2-f  5a;-l  and  3ar^-x2__2a;-5. 

32.  a&  +  ccZ  -\-aG-{-bd  and  a6  +  cd  —  ac  —  bd. 
*.33.  2a3-5a2_6a  +  4  and  4^^^  lOa^- 12a-8. 

Find  the  product  of  the  following : 

34.  X  — 3,  a;  +  4,  andic  — 7. 

35.  a-{-b,  a^-ab  +  b%  and  a^  —  b^. 

36.  2m- 1,  3m +  4,  and  6m -5. 

37.  a7  +  l,  3aj--2,  and3a;2__^_2. 

38.  aj'4-a5  +  l,a^-a;  +  l,anda;^-ar^+l. 


34  ALGEBRA. 

39.  a  +  b,  a  —  b,  a^  +  b\  Siud  a'^  +  b*. 

40.  mH- 1,  m— 1,  m  + 2,  and  m  —  2. 

41.  2i»— 1,  3a;+2,  4x  — 3,  and  5a;  +  4. 

42.  a-{-b,  a-b,  a-[-2b,  and  a? -2a''b-ab^ -\-2W. 

83.  The  product  of  two  or  more  polynomials  may  be  indi- 
cated  by  enclosing  each  of  them  in  a  parenthesis,  and  writing 
them  one  after  the  other. 

Thus,  the  product  of  a;  +  2,  »  —  3,  and  2 a?  —  7  is  indicated 

by  * 

(a;  +  2)(a;-3)(2a;-7). 

Similarly,  the  expression  (a+6  +  c)^  indicates  that  a -f  6-+- c 
is  to  be  multiplied  by  itself  (Art.  13). 

When  the  operations  indicated  are  performed,  the  expres- 
sion is  said  to  be  expanded  or  simplified. 

EXAMPLES. 
1.    Simplify  the  expression  {a  —  2xy—2{x-\-^a){a  —  x). 
To  simplify  the  expression,  we  should  expand  {a—2xY 
and  2  (a;  +  3  a)  (a  —  a?) ,  and  subtract  the  second  result  from 
the  first. 

{a—  2xY=  o?  —  A:  ax -\- 4:0? 

2{x  +  ^a){a-x)  =6a^  -  4:ax-2x^ 
Subtracting  the  second  result  from  the  first,  we  have 
a^—4:ax-\-4:X^  —  6a^-i-4:ax-\-2o(?=6x^—6a^y  Arts. 
Simplify  the  following : 

/2:  (a-hb  +  c-^dy, 

\Z.    {a-b)(c-d)  -\-(a-c)(b-d). 

4.  (2a;  -  3)2 +(1- a;)  (3a; -9). 

5.  (a-\-b-hcy-(a-b-\-cy. 

6.  (2a-56)2-4(a-26)(a-36). 


MULTIPLICATION.  86 

7.  (a  -  by  (a  +  by. 

8.  {l+x){l-j-x*){l-x-^x'-o^). 

9.  (l+ay-{l-a){l  -\-a'). 

10.  lx-{2y  +  ^z)]lx-{2y-3z)2.  '    ' 

11.  (x-^y){a^-f)[x^-y{x-y)}. 

12.  {a-^b){b-\-c)  -  {c-\-d){d-{-a)  -  {a -\- c)  {b  -  d) . 

13.  (a  4-  6  4-  c)2  +  (a  -  6  -  c)2+  (6  -  c-a)^-}-  (c-a-6)^ 

14.  (a-6)(6-c)  +  (6-c)(c-a)-f(c-a)(a-6). 

15.  if(ic-22/)+y(2/-22;)  +z(;2  -  2  a;)  -  (x- 2/-2;)2. 

16.  a;  (a; +  1)  (a; +  2)  (a;  4-3)  4- 1  -  (ar  4- 3x4- 1)'. 

17.  (a  4-  6  4-  c)2  -  (a  -  6  -  cy+  {b-c-ay-  (c-a-by. 

18.  [(m  4-  2ny  -  {27n  -  n)^] [(2m  4-  ny  -  (m  -  2n)2]. 

19.  (x-\-y  +  zy-(x^-\-f-j-z^)-3(y-j-z)(z-hx){x-^y), 

84.  Since  (  4-  a)  (  4-  5)  =  a6,  and  (  —  a)  (  —  6)  =  a6,  it  fol- 
lows that  in  the  indicated  product  of  two  factors  all  the  signs 
of  both  factors  may  be  changed  without  altering  the  value  of 
the  expression.     Thus, 

(a  —  6)  (c  —  d)  is  equal  to  {b  —  a){d  —  c) , 

Similarly,  we  may  show  that  in  the  indicated  product  of 
any  number  of  factors,  the  signs  of  any  even  number  of  factors 
may  be  changed  without  altenng  the  value  of  the  expression. 

Thus,  (ci  —  &)  (c  —  d)  (e  — /) ,  by  changing  the  signs  of  the 
second  and  third  factors,  may  be  written  in  the  equivalent 
form  (<x  —  b){d  —  c)  (/—  e) . 


36  ALGEBRA. 


VI.    DIVISION. 

85.  Division,  in  Algebra,  is  the  process  of  finding  one  of 
two  factors,  when  their  product  and  the  other  factor  are 
given. 

Hence,  Division  is  the  converse  of  Multiplication. 

Thus,  the  division  of  14 ab  by  7  a,  which  is  expressed 

(Art.   15),  signifies  that  we  are  to  find  a  quantity  which, 
when  multiplied  by  7  a,  will  produce  14:  ab. 

86.  The  Dividend  is  the  product  of  the  factors. 
The  Divisor     is  the  given  factor. 

The  Quotient   is  the  required  factor. 

87.  Since  the  dividend  is  the  product  of  the  divisor  and 
quotient,  it  follows,  from  Art.  77,  that : 

If  the  divisor  is  + ,  and  the  quotient  is  + ,  the  dividend  is  -f . 
If  the  divisor  is  — ,  and  the  quotient  is  + ,  the  dividend  is  — . 
If  the  divisor  is  + ,  and  the  quotient  is  — ,  the  dividend  is  — . 
If  the  divisor  is  — ,  and  the  quotient  is  — ,  the  dividend  is  + . 

In  other  words,  if  the  dividend  and  divisor  are  both  +,  or 
both  — ,  the  quotient  is  +  ;  and  if  the  dividend  and  divisor 
are  one  -|-,  and  the  other  — ,  the  quotient  is  — .  Hence,  in 
Division  as  in  Multiplication, 

Like  signs  produce  + ,  and  unlike  signs  produce  — . 

88.  Required  the  quotient  of  14  ab  divided  by  7  a. 

By  Art  85,  we  are  to  find  a  quantity  which,  when  multi- 
plied by  7  a,  will  produce  14  ab.  That  quantity  is  evidently 
2  b ;  hence 

Uab 


=  26. 


7a 


That  is,  the  coefficient  of  the  quotient  is  the  coefficient  of  th§ 
dividend ^  divided  by  the  coefficient  of  the  divisor, 

J 


DIVISION.  37 

89.  Required  the  quotient  of  a^  divided  by  a^. 

We  are  to  find  a  quantity  which,  when  multiplied  b}"  ^^, 
will  produce  a^.     That  quantity  is  evidently  o?  ;  hence 

-  =  a\ 

That  is,  the  exponent  of  a  letter  in  the  quotient  is  equal  to 
its  exponent  in  the  dividend  minus  its  exponent  in  the  divisor. 

For  example,  —  =  a'"", 

a'* 

DIVISION    OF    MONOMIALS. 

90.  We  derive  from  Arts.  87,  88,  and  89  the  following 
rule  for  the  division  of  monomials : 

To  the  quotient  of  the  coefficients  annex  the  literal  quantities^ 
giving  to  each  letter  an  exponent  equal  to  its  exponent  in  the 
dividend  minus  its  exponent  in  the  divisor.  Make  the  quotient 
-h  when  the  dividend  and  divisor  have  like  signs,  and  —  when 
they  have  unlike  signs. 

EXAMPLES. 
1.    Divide  54a^by  -da\ 

By  the  rule,     ^M-  =  - 6a'-'  =  - 6a\.  Ans. 
^  -da* 


2.    Divide  -  2a^b'cd^  by  abd' 
-2a^b-cd^ 


abd' 


—  2a-6c,  Ans. 


Note.  A  literal  quantity  having  the  same  exponent  in  the  dividend 
and  divisor,  as  o?*  in  .Ex.  2,  is  canceled  by  the  operation  of  division,  and 
does  not  appear  in  the  quotient. 

3.    Divide  —  91  x^Y^^  by  —  13a;"2/V. 


38  ALGEBRA. 

Divide  the  following : 

4.  84  by  -12.  14.  -  ISa^fz  by  da^z. 

5.  -  343  by  7.  15.  -  65a«6V^  by  -  5a6V. 

.      6.   -824  by  -18.  16.  72  m%  by  -12m2. 

-  7.  444  by  -  37.  17.  12a;Y  ^J  ^^Y- 
"     8.  12a''  by  4a.  18.   -  18a™5  by  ^ah. 

-  9.   -  a^c  by  om.  19.   -  144c^d^e«  by  -  ^^cH^e, 

10.  2  m^7i*  by  -  mn^.  20.  -  3  a"'+2  by  a'"+^ 

11.  -8icyby-4a^.  21.  a'"+''&™+"  by  a'"5\ 

12.  30a^6^  by  hd'h,  22.  -  Ola^y^j^  ^y  _l3a^2/^. 

13.  14m^n'*  by  —  7mn-^.  23.  18m'Wj)^  b}^  —  2m^n2/. 

DIVISION    OF    POLYNOMIALS    BY    MONOMIALS. 

91.   The  operation  being  simply  the  converse  of  Art.  81, 
we  have  the  following  rule  : 

Divide  each  term  of  the  dividend  by  the  divisor^  and  conned 
the  results  with  their  proper  signs. 

EXAMPLES. 

1 .  Divide  9  a^^  -^  6  a^c  +  1 2  a^bc  by  -  3  a\ 

By  the  rule, 

9a35_6a^c+12a^6c^_3^  ,^_^        ^^^^ 

-3a' 

Divide  the  following : 

2.  8a^6c+16a'6c-4a''62by  4-a2c. 

3.  9a;^+27ar^-2l£c2by  -Sx^. 

4.  30a3-75a^6by  15  a^ 

5.  2a^yh  -Uxy-z"  by  -  2  xy'z. 


DIVISION.  39 

6.  5  a%G-~  5 aly^c  -\-  5  ah(?  by  —  5 ahc, 

7.  4x^-8a^-14i»^  +  2a;*-6a:^by  2«3. 

8.  - 12  a^¥  -  30aPl^  -u  108 a"6"  by  -  6  a"*6". 

9.  20a;^-12a^-28a;by  4.T. 

10.  -  a'b'c  -  ab'c'  +  a^bc^  by  -  abc. 

11.  da'bc-sJb  +  lSa^bchy  -Sab. 
12.15  x'^y'^z'  -  35 af^^y^z  by  5  ic"*?/"^. 
13.  20 a^6c  +  15  abd^  - 10 a'b  by  -  5  ab. 

DIVISION  OF  POLYNOMIALS  BY  POLYNOMIALS. 

92.  Required  the  quotient  of  12  +  lOar^  —  11  x  —  2).r' 
divided  by  2  ic^  —  4  —  3  a;. 

Arranging  both  dividend  and  divisor  according  to  the 
descending  powers  of  x  (Art.  37),  we  are  to  find  a  quantity 
which,  when  multiplied  b}'  the  divisor,  2ar^  — 3  a;  — 4,  will 
produce  IO-t^- 21ar^- lla;+ 12. 

It  is  evident,  from  Art.  82,  that  the  term  containing  the 
highest  power  of  x  in  the  product,  is  the  product  of  the 
terais  containing  the  highest  powers  of  x  in  the  factors. 
Hence  1  Oaf'  is  the  product  of  2x'  and  the  term  containing 
the  highest  power  of  x  in  the  quotient.  Therefore  the  term 
containing  the  highest  power  of  x  in  the  quotient  is  10  a? 
divided  by  2af^,  or  5x. 

Multiplying  the  divisor  by  5  a;,  we  have  the  product 
lOa:^  — 15a;^— 20a;;  which,  when  subtracted  from  the  divi- 
dend, leaves  the  remainder  —  6a;^  +  9  a;  -f  12. 

This  remainder  is  the  product  of  the  divisor  by  the  rest 
of  the  quotient ;  hence,  to  obtain  the  next  term  of  the  quo- 
tient, we  proceed  as  before,  regarding  — 6a,'^  +  9a;  +  12  as  a 
new  dividend.  Dividing  the  term  containing  the  highest 
power  of  a;,  —  6a;^,  by  the  term  containing  the  highest  powei 


iO  ALGEBRA. 

*)f  X  in  the  divisor,  2x^,  we  have  —  3  as  the  second  term  of 
the  quotient. 

Multiplying  the  divisor  by  —  3,  we  have  —  6  it-^ +  9  a; +  12; 
which,  when  subtracted  from  the  second  dividend,  leaves  no 
remainder.     Hence  5ic  — 3  is  the  required  quotient. 

It  is  customary  to  arrange  the  work  as  follows  : 

2  ar  —  3  a:  —  4 ,  Divisor. 


10a^-21flr^_iia;+12- 
10a;3-15a^-20a; 


5  a;  —3,  Quotient. 


6x^+    9a;  +  12 
6xP+   9a;+12 


Note.  We  might  have  solved  the  example  by  arranging  the  divi- 
dend and  divisor  according  to  the  ascending  powers  of  x,  in  which  case 
the  quotient  would  have  appeared  in  the  form  —  3  +  5  a:. 

93.  From  Art.  92,  we  derive  the  following  rule  for  the 
division  of  polynomials : 

Arrange  both  dividend  and  divisor  in  the  same  order  of 
powers  of  some  common  letter. 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor^  giving  the  first  term  of  the  quotient. 

Multiply  the  whole  divisor  by  this  term,  and  subtract  the 
product  from  the  dividend,  arranging  the  remainder  in  the 
same  order  of  powers  as  the  dividend  and  divisor. 

Regard  the  remainder  as  a  new  dividend,  and  proceed  as 
before;  continuing  until  there  is  no  remainder. 

Note.  The  work  may  be  verified  by  multiplying  the  quotient  by 
the  divisor,  which  should  of  course  give  the  dividend. 

EXAMPLES. 
1.  Divide  21s(fy^  -  22xy  -  8  by  3a;y  -  4. 


21xy-22xy-8 

Sxy  —  4: 

21x'f-28xy 

lxy-\-2,  Ans 

6xy-8 
6xy  —  8 

DIVISION.  41 

2.    T>iyide8-{-18x^-66x'hy  -ex^  +  4:-\-8x. 

Arranging  according  to  the  ascending  powers  of  x, 

8_56a^+18aj^   4:  +  8x-6a^ 
8+16a;  -12a^ 


2  —  4a;  — 3a^,  Ans. 


-16x  -Ux'  +  18x'^ 

-16a;-32a^  +  24fl^ 

-12a^-24a;3_^lg^4 

-12a^-24g;^4-18a;^ 

3.   Divide  9ab^  +  a^-9b^-  bo?h  by  Sfe^  +  a^-  2db, 
Arranging  according  to  the  descending  powers  of  a, 

a^-2a'b  +  Sab^ 


Divide  the  following : 

4.  6a;2_  ^_  35  |3y  3^^  Y^ 

5.  2  -  3aa;  -  2aV  by  1  -  2ax. 

6.  a2-4a6  +  4?)2by  a-26. 

7.  59a;-56-15ar^by  3a;-7. 

8.  Sb^  +  Sab^-Aa'b-^a^hyb  +  a, 

9.  2  a^x  —  2  aa:^  by  ax  —  a^. 

10.  18a^-5a;  +  l  by  6a:2_|_2a;_i. 

11.  8m3  +  35-36mby  5-|-2m. 

12.  27a^  +  2/^  by  3x4-2/. 

13.  16m^-l  by  2m-l. 

14.  a^-b'-i-c^-2aGhy  a-i-b-c. 

15.  8 a^ -\- 36 a-b-\-54:ab'-+- 27 b^hy  2a -{-Sb. 

16.  x'  +  2/  -{-xhf  by  x'  -^y'  +  xy. 


42  ALGEBRA. 

17.  2x*-19a^  +  9  by  2ar^  +  6a;2_a;-3. 

__  18.  8  m^  -\- S  n^  —  4:  mhi  —  6  mn^  by  2m  —  n. 

19.  4a;^-8a^-6a?2  +  24  by  2a;-4. 

20.  23a^ -48  4- 6rc*- 2a;- 31  a^  by  6  4-3.^2-50?. 
'21.  4a^  +  27-a«by  9-3a3  4-4a2  +  2a4_6a. 

22.  x^  —  dsc^  —  exy-y^hy  a^-^Sx  +  y. 

23.  a8-816n)y  a2  +  36. 

24.  a^  —  2/"  +  2^/2;  —  2;-  by  a;  4-  2/  —  2!. 

25.  3a;4-14a?2  4-8by  a;-2. 

26.  y^  +  x'^y  hy  x  +  y. 

27.  15m^4-50m2  4-15-32m-32m3by  3m2  4-5-4m. 

28.  l-h4:X^  +  Sx^hy  (x  +  iy. 

29.  21a''-216«by  la-lh. 

30.  64a;^4-l  by  8a;2-4a;4-l. 

31.  50a5  4-9a;*4-24-67a;2i^y  ^^^2_g_ 

32.  a;^ 4- 2/* ~  ^^if  ~  4a;^2/  +  Ga^^^/^  by  x^  -\-y'^  —  ^Xfy. 

33.  aj^-4a^4-2.'«-4-4a;4-l  by  (a;-l)2-2. 

34.  9a!^4-4?/^-37a;y  by  3a^-2/4-5a;2/. 

35.  a*  4- a'^' 4- 25 6^  by  (a- 6)  (a- 56)  +  3a&. 
_36.  3a^4-4a;4-6ar'-lla;^-4by  3a;2-4. 

37.  6ar5  4-15ar^4-51a;-18  by  2a;3-4a^4-7a?-2. 

38.  2a;*-lla;-4a;2_i2_3a;3i3y  4_^2a.-2  4-a;. 

39.  m*  —  48  —  17m3  4-  52m  4-  12m2  by  m  —  2  4-  w^ 
.  40.  a;"+^  4-  a;"?/  —  xy'^  —  y'^-^^  by  x^  —  y^. 

41.  x'y  —  x]f  hy  y?  ■\- if"  -\-  xy'^  4-  a^^?/* 

42.  x^  —  ^^-x-^\>y  x?-\-'lx-\-^. 

43.  2a''  4-  SSa^fts  _  49  6^  -  la^y"  -^a%  by  2a^-5ab  -751 


DIVISION. 


43 


45.  2a^-6y^-V2z^-{-xy-2xz-i-17yzhy  2x  +  4:Z-Sy. 

46.  a^n  _  52«  _j_  2  ^-c'"  —  c^'"  by  a'^  +  &"*  -  c". 

47.  a;«_  1  _  6»^-3a^  by  -20.-2 -aj  +  a^-l. 

48.  12  a^-14a*&  +  ^Oa'V  -  a-^^ _  8a6^+  46^ 

by  6a^- 4 cr6- 3 aft^H- 2 6^. 

The  operation  of  division  may  be  abridged  in  certain  cases 
by  the  use  of  parentheses. 

49.  Divide  (a^  +  ab)x^  +  (2ac  +  6c  +  ad)x  +  c{c  +  d) 

by  ax  -{-  c. 


(a2+  ab)os^  +  (2ac  +  be  -\-  ad)x  +  c{c  +  d) 
(a^-\-ab)a^  +(  ac  +  bc         )x 


ax-\-c 


(   ac 
(   ac 


-\-ad)x-\-c{c-{-d) 
+  ad)x-\-c(c  +  d) 


{a+b)x+{c+d), 
Ans, 


Divide  the  following : 

50.  a^  +  (a-{-b-i- c)x-  +  («6  +  bc  +  ca)x  +  abc 

by  fl^+(&+c)ic  +  6c. 

51.  (6  +  c)a2H-(6-4-36c  +  c2)a  +  6c(6  +  c)  by  a  +  &+c. 

52.  (x+yy-^(x  +  y)  +  6hy  {x  +  y)-2. 

53.  (a  +  6)3_j-i  by  (a  +  5)  +  l. 

54.  ar^  +  (a  4-  6  —  c)x^  +  (a&  —  6c  —  ca)a;  —  a6c 

by  x'^-{-{b  —  c)x  —  be. 

65.   (m-7i)*-2(m-n)2  +  l 

by  (m  —  n)2  —  2 (m  —  n)  +  1 . 

56.  if^-f- (a  —  6  +  c)ar  +  (ac  —  ab  —  bc)x  —  abc  by  a;  +  c. 

57.  a-^  +  (3  -  6).^•^  +  (c  -  3  6  -  2) a^  +  (2  6  +  3  c)a;  -  2 c 

by  or  +  3  x  —  2. 

58.  a-(6  +  c)  +  a(62  +  6o  +  c')  -  6c(6  +  <■)  by  a  +  6+  c. 


44  ALGEBRA. 


VII.  FORMULAE. 

94.  A  Formula  is  the  algebraic  expression  of  a  general 
rule. 

95.  The    following   results   are   of   great   importance   in 
abridging  algebraic  operations : 


a  +b 

a-b 

a +6 

a  -\-b 

a-b 

a-b 

a'-^-ab 

0?—  ab 

a^-\-ab 

ab    +62 

-ab    +62 
a2_2a6  +  62 

-ab- 

-b' 

a:'+2ab  +  b^ 

a' 

-b' 

In  the  first  case,  we  have  (a-\-  by  =  a'^  -\-  2 ab  -j-  b^.        (1) 

That  is,  the  square  of  the  sum  of  two  quantities  is  equal  to 
the  square  of  the  first,  plus  twice  the  product  of  the  two,  plus 
the  square  of  the  second. 

In  the  second  case,  we  have  (a  —  by  =  a^—2ab-\-  6^.    (2) 

That  is,  the  square  of  the  difference  of  two  quantities  is 
equal  to  the  square  of  the  first,  minus  twice  the  product  of  the 
two,  plus  the  square  of  the  second. 

In  the  third  case,  we  have  (a  +  5)  (a  —  6)  =a^  —  b^.      (3) 

That  is,  the  product  of  the  sum  and  differeyice  of  two  quan- 
tities is  equal  to  the  difference  of  their  squares. 

EXAMPLES. 

96.    1.  Square  3o+26c. 

The  square  of  the  first  term  is  9a^,  twice  the  product  of 
the  terms  is  12  abc,  and  the  square  of  the  second  term  is 
4  6^c^.     Hence,  b}^  formula  (1), 

(3  a  +  26c)2  =9a2+ 12  a6c  + 46V. 


FORMULA.  45 

Note.  The  following  rule  for  the  square  of  a  monomial  is  evident 
from  the  above : 

Square  the  coefficient,  and  multiply  the  exponent  of  each  letter  by  2. 
Thus,  the  square  of  ba%  is  250*62. 

2.  Square  4fl7  — 5. 

By  formula  (2),   (4a; -5)^  =  163^- 40a;  +  25,  Ans, 

3.  Multiply  6  a^  +  &  by  6  a2  -  6. 

By  formula  (3) ,   {<c>a- +  b)  (6a^  -  6)  =  36  a*  -  b^  Ans. 

Write  by  inspection  the  values  of  the  following : 

4.  (a; -4)2.  16.   (3^3  .^13)2^ 

5.  (S  +  ay.  17.   {ea'-b'cy. 

Q.  (a; 4- 3) (a; -3).  18.  {oa+7b'){oa^7ly'). 

7.  (3a +  5)^  19.  (13a&  +  5ac)2. 

8.  (2a;  +  l)(2a;~l).  20.  (a^ -^5x){a^ - 5x), 

9.  (7-2a;)2.  21.  (l-Uxyzy, 

10.  {2m  +  Sny.  22.   {4.a^ -\-3f)(4:x'-3f). 

11.  {4.ab-xy.  23.   {10a^-\-9a^y. 

12.  (5  + 7a;)  (5 -7a;).  24.   (4a^-56«)^. 

13.  (x'-fy.  25.   («"*  +  «") (a"* -a"). 

14.  (3a;+ll)(3a;-ll).  26.   (la^  +  Uxy. 

15.  (a;V  +  4)2.  27.  (5a'«-a'*)^ 

28.    Multiply  a  +  6  +  cbya  +  &—  c. 

(a  +  6  +  c)(a  +  6-c)  =  [(a  +  6)  +  c][(a  +  6)-c] 

=  (a  4-  6)'  -  cS      by  formula  (3) 
=  a2  +  2a6  +  62_c2,  ^^s. 


46  ALGEBRA. 

29.    Multiply  a  +  b  —  chya  —  b  +  c. 

{a-hb-c){a-b-\-c)  =  la  +  {b-c)'][a-(b-c)^ 
=  a'-(b-cy 
=  a'-(b'-2bc  +  c') 
=  a'-b^-i-2bc-c',  Ans. 

Expand  the  following : 

30.  {x-\-y-{-z)(x-y  +  z).         32.    (1  +  a- 5)  (1- a  +  &). 

31.  {x-\-y  +  z){x-y-z).         33.   (x^ -^x+1)  {x^ -x-1). 

34.  (a-j-b  —  c)(a-b-c). 

35.  {a^  +  2a  +  l)(a'-2a  +  l). 

36.  {x^  +  2x-S){xP~2x-{-S), 

37.  (m^  +  ^^  +  ^^)  (^^^  —  "^^  +  ^^)  • 


97.    We  find  by  multiplication  : 

X  +5 

.^•  -5 

X  +3 

X  —n 

si^  +  6x 

a^  —  5x 

+  3ar+15 

-3i»  +  15 

a^  +  8aj+15 

a^-8a;+15 

a;  +5 

X  —5 

X  -3 

X  +3 

ar^  +  5a; 

af  —  5x 

-3a;-15 

+  3a;-15 

a^^2x-15  '  a^-2a;-15 

We  observe  in  these  products  the  following  laws  : 

I.  The  coefficient  of  x  is  the  algebraic  sum  of  the  numbei^ 
in  the  factors. 

II.  The  last  term  is  the  product  of  the  numbers. 

By  aid  of  the  above  laws  the  product  of  two  binomials  of 
the  form  x  +  «,  x-\-b  may  be  written  by  inspection. 


FORMULAE.  47 

1.  Required  the  value  of  (x  —  8)  (x  +  5) . 

The  coefficient  of  «  is  —  3  ;  and  the  last  term  is  —40. 
Hence,  (^x-8){x  +  5)  =  af -3x-4:0,  Ans. 

EXAMPLES. 
Write  by  inspection  the  values  of  the  following : 

2.  (x-{-7){x  +  o),  10.    (x  +  9){x-o). 

3.  (a;-3)(a;-4).  11.    {x-8)(x-d). 

4.  {x  +  8){x-2).  12.    (x-\-4:m){x  +  Q7n), 

5.  {x-S){x-\-l).  13.    (x-5a)(it'  +  a). 

6.  {x-5){x  +  6).  14.    (a4-6)(a-46). 

7.  (.'»  +  l)(a;-|-12).  15.    {a  +  i)b)(a  +  Sb). 

8.  (a;-7)(a;-f-2).  16.    (a.-^  -  3)  (a^  -  7) . 

9.  {x-8)(x-6).  17.    (aj3  +  2a)(ar'^-6a). 

98.    The  following  results  may  be  verified  by  division : 

(1)  ^!LZL^  =  a-b.  (3)  ^L±^  =  a'-ab  +  bK 

(2)  tzL^  =  a  +  b.  (4)  ^^^'  =  a*+a64-6^ 

a— &  a— 6 

Formulae  (3)  and  (4)  may  be  stated  in  words  as  follows  : 

If  the  sum  of  the  cubes  of  two  quantities  be  divided  by  the 
sum  of  the  quantities,  the  quotient  is  equal  to  the  square  of 
the  first  quantity^  minus  the  product  of  the  two,  plus  the  square 
of  the  second. 

If  the  difference  of  the  cubes  of  two  quantities  be  divided  by 
the  difference  of  the  quantities,  the  quotient  is  equal  to  the 
square  of  the  first  quantity,  plus  the  product  of  the  two,  plus 
the  square  of  the  second. 


48  ALGEBRA. 

EXAMPLES. 

1.  Divide  36?/V  -  9  by  ^yz"  +  3. 

By  formula  (1) ,       — ^  „  ~     =  62/2^  —  3,   Ans. 
^  ^  62/22-1-3 

2.  Divide  1  +  Sa^  by  1  +  2a. 

By  formula  (3) ,       LLM  =  1  -  2  a  +  4  aS  Ans. 
^  ^  ^        l+2a 

3.  Divide  27a^-63by  3a-6. 

By  formula  (4) ,       ^^^'"^  =  9  a^  +  3  a6  +  5^,  ^715. 


EXAMPLES. 
Write  by  inspection  the  values  of  the  following : 


4. 

a^-81 
a;-9 

9. 

27  +  0^ 
3  +  a; 

14. 

x'-f 
x^-f 

5. 

25-16a2 
5  +  4a 

10. 

a^-16aj2 
a^  +  4a; 

15. 

^■\-xf 

6. 

0^+1 

x+1 

11. 

ajs_64 
a;-4 

16. 

49a2-121&^ 

7. 

1  — m 

12. 

l-2m 

17. 

64  m^  +  n^ 

8. 

a«-8 
a-2 

13. 

a3+343 
a  +  7 

18. 

x-^5y 

Divide  the  following : 

19.  27aj»2/^-642^by  3a;2/-4«. 

20.  25  a^  -  81  b^c"^  by  5  a^  _  9  6c». 

21.  343  +  125a;^y'lt)y  7  +  5a;2/. 


FORMULAE.  49 


22.  Um^-216n^hy  Am-6n\ 

23.  729xy+5122«by  9aT^2/  +  822. 

).    By  actual  division  we  obtain  : 


a-{-b 
a'-b* 


=  a?-  d'b  +  ab^  -  b\ 
=  a?-^a'b  +  ab''-{-W. 


a  —  b 

=  a*-  a%  4-  a'b^  -  aW  +  b\ 


«*  +  &'_.4 


a  +  6 
a  —  b 


=  a^  +  o?b  +  a?l^-\-dt^-\-b^\  etc. 


In  these  results  we  observe  the  following  laws : 

I.  The  number  of  terms  is  the  same  as  the  exponent  of  a 
in  the  dividend. 

II.  The  exponent  of  a  in  the  first  term  is  less  by  1  than 
the  exponent  of  a  in  the  dividend,  and  decreases  by  1  in 
each  succeeding  term. 

III.  The  exponent  of  b  in  the  second  term  is  1,  and 
increases  by  1  in  each  succeeding  term. 

IV.  The  terms  are  all  positive  when  the  divisor  is  a  —  6, 
and  are  alternately  positive  and  negative  when  the  divisor 
is  a  +  6. 

100.  In  connection  with  Art.  99,  the  following  principles 
are  of  great  importance  : 

If  n  is  any  whole  number, 

(1)  a**  +  6"  is  divisible  by  a-\-b  if  n  is  odd,  and  by  neither 
a-\-b  nor  a  —  b  if  n  is  even. 

(2)  a"  —  6"  is  divisible  by  a  —  b  if  n  is  odd,  and  by  both 
a-{-b  and  a  —  b  if  n  is  even. 


50  ALGEBKA. 

EXAMPLES. 
101.   1.  Divide  cH  -  H'  hy  a-b. 
Applying  the  laws  of  Art.  99,  we  have 

^Sll.  =  a«  +  a'b  +  a'b'  +  a'b^  +  a'b'  +  ab'  +  b',  Ans, 
a  —  b 

2.    Divide  a;^  -  81  by  a;  +  3. 
Since  81  =  3^,  we  have 

^^~^^  =  a^-3a;^  +  3'a;-3^  =  a^-3a^  +  9a;-27,  Ans, 
fl;  +  3 

Write  by  inspection  the  values  of  the  following  : 

3.  «°-»''.  8. 
a  —  b 

4.  ^:^.  9. 
x  +  y 

5.  "^'  +  <  10. 

6.  ^'-<  11. 


7.   i-Hi^.  12. 

l-x 


Divide  the  following : 

18.  m*-16n«by  m-27i2.  20.    ^2d -^b^  hy  2a  +  b. 

19.  Q^^i^T^hy  x  —  yz.  21.    m*- 243n'' by  m  — 3». 

22.    256x^-2/«by  4a;  +  y^ 


x' 

-16 

X 

-2 

1- 

-a^ 

1 

—  a 

a^ 

+  1 

a 

+  1 

1- 

-n« 

1 

—  n 

03^ 

-81 

13. 

aj«-32 
0^-2 

14 

a«-64 

a  +  2 

16. 

ai«  +  &^' 

a2  +  62 

16. 

aj^-128 

x-2 

17. 

x^  +  243 

a;  4-3 

FACTORING.  61 


VIII.  FACTORING. 

102.  Factoring  is  the  process  of  resolving  a  quantity  into 
its  factors.     (Art.  11.) 

103.  The  factoring  of  monomials  may  be  perfoi-med  by 
inspection ;  thus, 

12a^6-c=2.2.3aaa56c. 

A  polynomial  is  not  always  factorable ;  but  there  are 
certain  forms  which  can  always  be  factored,  the  more 
important  of  which  will  be  considered  in  the  succeeding 
articles. 

Case  I. 

104.  When  the  terms  of  the  polynomial  have  a  common 
monomial  factor. 

1.  Factor  a^  4-3  a. 

Each  term  contains  the  monomial  factor  a. 
Dividing  the  expression  by  a,  we  have  a^-f-S.     Hence, 
a^-f3a  =  a(a^4-3),  Ans. 

2.  Factor  Uxy*  — 35  a^/. 

Uxy^-35a^y^=7xy\2y'--5ay'),  Ans. 

EXAMPLES. 
Factor  the  following : 

3.  ar  +  6x.  8.  5a;=^-f  10a.-2  + 15a;. 

4.  Sm^-12m\  9.  a' -2a^  +  Sa^ -a\ 

5.  16a^-12a.  10.  SQx^y -GOx'y* -84:xy. 

6.  27c*cC^-\-d(^d.  11.  21m^n-\-35mn^-Umn. 

7.  60mV-12m3.  12.  SAx'f -UOx^y^-j-lOx^- 


62  ALGEBRA. 

13.  Factor  the  sum  of  54a^6^  -  72  aV,  and  -  90  a^d. 

14.  Factor  the  sum  of  96c*c?^  1200^^^  and  -  U4.(fd\ 

Case  II. 

105.  When  the  polynomial  consists  of  four  terms,  of  which 
the  first  two  and  the  last  two  have  a  common  binomial  factor, 

1.  Factor  am  —  hm  +  an  —  hn. 

Factoring  the  first  two  and  last  two  terms  as  in  Case  I, 
we  have 

m{a  —  h)  -\-  n{a  —  h) . 

Each  term  now  contains  the  binomial  factor  a~b.  Divid- 
ing the  expression  hj  a  —  b,  we  obtain  7n  +  n.     Hence, 

am  —  bm-\-an  —  bn  =  (^a  —  b){m-\-  n) ,  Ans. 

2.  Factor  am  —  bm  —  an  +  bn. 

am  —  bm  —  an-\-bn  =  am  —  bm  —  {an  —  bn) 
=  m(a  —  b)—n(a  —  b) 
=  (a  —  b)  (m  —  n) ,  Ans. 

Note.  If  the  third  term  is  negative,  as  in  Ex,  2,  it  is  convenient, 
before  factoring,  to  enclose  the  last  two  terms  in  a  parenthesis  preceded 
by  a  —  sign. 

EXAMPLES. 
Factor  the  following : 

3.  ab -\- bx -\-  ay  -{-  xy.  8.  a^  —  a^b  —  ab'^  +  W. 

4.  ac  —  cm  -\-ad  —  dm.  9.  o^  +  ax  —  bx  —  ab. 

5.  x^-\-2x  —  xy  —  2y.  10.  mx^  —  my^  +  ?ix^  —  ny'^. 

6.  Q(?  —  ax  —  bx-\-ab.  11.  x^  -\- x^  -\- x -\- \ . 

7.  o?-a%-\-ab''-b\  12.  ^x^ ^\x? -9x-^. 

13.  ^cx—\2cy^2dx  —  Zdy. 

14.  6n-21m2n-8m  +  28m^- 


FACTORmG.  53 

106.  If  a  quantity  can  be  resolved  into  two  equal  factors, 
it  is  said  to  be  imperfect  square^  and  one  of  the  equal  factors 
is  called  its  square  root. 

Thus,  since  9  a^6^  equals  3a-6  X  3a^6,  it  is  a  perfect  square, 
and  3  a^h  is  its  square  root. 

Note.  9  a^b-  also  equals  —  3  a^i  x  —  3  a%,  so  that  its  square  root  is 
either  3  d-b  or  —  3  a-b.  In  the  examples  in  this  chapter  we  shall  con- 
sider the  positwe  square  root  only. 

107.  The  following  rale  for  extracting  the  square  root  of 
a  monomial  is  evident  from  Art.  106  : 

Extract  the  square  root  of  the  coefficient^  and  divide  the 
exponent  of  each  letter  hy  2. 

For  example,  the  square  root  of  ^ba^^fz^  is  hs^y^z. 

108.  It  follows  from  Art.  05  that  a  trinomial  is  a  perfect 
square  when  its  first  and  last  terms  are  perfect  squares  and 
positive,  and  the  second  term  is  twice  the  product  of  their 
square  roots. 

Thus,  4a^  —  12£c?/  +  9?/^  is  a  perfect  square. 

109.  To  find  the  square  root  of  a  perfect  trinomial  square, 
We  take  the  converse  of  the  rules  of  Art.  95  : 

Extract  the  square  roots  of  the  first  and  last  terms ^  and 
wnnect  the  results  by  the  sign  of  the  second  term. 

Thus,  let  it  be  required  to  find  the  square  root  of 
^x^-l2xy-\-^y-. 

The  square  root  of  the  first  term  is  2  a;,  and  of  the  last 
term  3  2/;  and  the  sign  of  the  second  term  is  — ,  Hence 
the  required  square  root  is 

2a;  — 3y, 


54  ALGEBRA. 

Case  III. 

110.   Wlien  a  trinomial  is  a  perfect  square  (Art.  108). 

1.  Factor  a^  +  2  ab' +  b\ 

By  Art.  109,  the  square  root  of  the  expression  is  a  +  b^. 
Hence, 

a2  4-  2  ab'  -\-b'=(a-i-  b')  (a  +  b') ,  or  (a  +  b'Y,  Ans. 

2.  Factor  4tx^ -  12xy  +  9y\ 

4:X^—12xy-{-dy^  =  {2x-3y){2x-3y) 
^{2x-3yy,  Ans. 

Note.    The   given   expression   may   be  written  di/^  —  12xi/-{-4x'^; 
whence, 

9j/2_  12x^4- 4^-2=  {Sy-2x){3r/-2x)  =  {Sy-2x)^; 
which  is  another  form  of  the  answer. 

EXAMPLES. 
Factor  the  following : 

3.  x^-\-2xy-}-y\  16.  36  m^  -  36  mn  +  d  nK 

4.  4  +  4m  +  m2.  17.  4:a^ -{-Uab -h  "i 21  b\ 

5.  a^- 14a; +  49.  18.  x^ -\-8x^ -\-16x\ 

6.  a^- 10 a  4- 25.  19.  a'b* -\- 18  ab'c -\- 81  c^.^ 
1,  2/^  +  22/  +  l.  20.  2bx--10xyz-{-4.^y-z^. 

8.  m2-2m+l.  21.  'do^ -66x^ -^121x\ 

9.  a;*+12a;2  +  36.  22.  da^ -{-60a''bG'd+100b^(^d\ 

10.  n«-20n3  +  100.  23.  64»«- 1600;^  + 100a;«. 

11.  icy+ 160^2/  + 64.  24.  la^l)' -{-h2a^b^  ^U^a^bK 

12.  l-l0ab  +  2ba'b\  25.  16a;*-120mna^+225mV. 

13.  16m'^-8am  +  a\  26.  {a-by -\-2{a-b)+l. 

14.  a^  +  2a^-{-a\  27.  {x-^yy-16{x-\-y)^64.. 

15.  x"^ - 4.x' +  4:x\  28.  {x'-xy  +  6(x'-x)  +  ^. 


FACTORIKG.  55 

Case  IY. 

111.  When  an  expression  is  the  difference  of  two  perfect 
squares. 

Comparing  with  the  third  case  of  Art.  95,  we  see  that 
such  an  expression  is  the  product  of  the  sum  and  difference 
of  two  quantities. 

Therefore,  to  obtain  the  factors,  we  take  the  converse  of 
the  rule  of  Art.  95  : 

Extract  the  square  root  of  the  first  term  and  of  the  last 
term;  add  the  results  for  one  factor,  and  subtract  the  second 
result  from  the  first  for  the  other. 

1.  Factor  36  a.-2- 49/. 

The  square  root  of  the  first  term  is  6  a;,  and  of  the  last 
term  ly.     Hence,  by  the  rule, 

36ar-49/  =  (6x-f  72/)(6ic-72/),  Ans, 

2.  Factor  (2  a; -3  2/)  2 -(a;-?/)  2. 
{2x-^yY-{x-yy 

=  [i2x-Zy)-\-{x-y)-][{2x--^y)-{x-y)-] 
=  {2x-3y  +  x-y){2x-3y-x-\-y) 
=  (Bx  —  4:y){x  —  2y),  Ans. 

EXAMPLES. 
Factor  the  following : 

3.  x'-y^  7.  9a.'2-16/.  11.  Adm^-100n\ 

4.  ar'-l.  8.  2oa'-b'.  12.  36x^-81y\ 

5.  4:-a\  9.  l-49a^/.  13.  64a2- 1216V. 

6.  9m=^-4.         10.  a'b^-c'd\  14.  lUxY-225z''. 


56  ALGEBRA. 

15.  (a-\-by~{c-\-dy.  19.  (x-cy-(y-dy. 

16.  (a-c)2-62.  20.  (a-3y-{h  +  2y. 

17.  m^-{x-yy.  21.  (2x  +  my -(x-7ny. 
IS.  m'-{m-iy.  22.  {3a-\-5y -{2a-3y. 

It  is  sometimes  possible  to  express  a  polynomial  in  the 
form  of  the  difference  of  two  perfect  squares,  when  it  may 
be  factored  by  the  rule  of  Case  IV. 

23.  Factor  2  m7i  +  m^  -  1  -f  7i\ 

The  expression  ma}'  be  written  m^  +  2  mn  +  n^  —  1,  which, 
by  Case  III.,  is  equivalent  to  (m-|-?i)^—  1.  Hence,  by  the 
rule, 

(m  +  ?i)^  —  1  =  (?7i  +  ?i  +  1)  (m  +  w  —  1) ,  A71S. 

24.  Factor  2xy -{- 1 -x^ -f. 

2xy  +  1  -  x"  -  y'-  =  1  -  x"  +  2xy  -  y'-  =  1-  (ay^-  2xy+y^) , 

By  Case  III.,  this  may  be  written  l—(x  —  yy.  Hence 
the  factors  are 

ll+{x-~y)^[l-(x-y)-]^(l-}-x-y)(l-x  +  y),  Ans. 

25.  Factor  2xy  +  W  -  x^  -  2ab  —  y"^  -\-  a\ 

2xy-\-b^-x^-2ab-y^-\-a' 

=^  a" -2ah  -{-b'' - x"  +  2xy  - y^ 

=  a'-2ab  +  b'-(i^-2xy-hy^) 

=  (a  -  6)2  _  (a;  _  2/)2,  by  Case  III. 

=  [(a-6)+(a.-2/)][(a-6)-(a;-y)] 

=  (a  —  6  +  a;  —  2/)(a  —  6  —  a;  +  2/),  Ans. 

Factor  the  following : 

26.  x'-^2xy  +  y^-A.  28.  a2-62  4-26c- c^. 

27.  a2-2a6  +  &'-c2.  29.  a^-ft^- 26c-c2. 


FACTORING.  57 

30.  c^  _  1  +  fZ2  -f-  2cd.  32.  4&  -  1  -  46-  +  4m^       - 

31.  9_a^_2/2_^2a:2/.  33.  4a' -{-b' -9d' -  4.ab. 

34.  •  a'  -  2am  -]-m' ~b' -2bn-  nK 

35.  x'-y'-{-c'-d''-2cx  +  2dy. 

36.  a2  -b'-j-  m-  -  ii'  +  2  am  +  2  6n. 

37.  a'-W  +  c'-d'-ir2(M-2bd. 

Case  V. 
112.  When  an  expression  is  a  trinomial  of  the  form  a^-\-  ax-{-b. 

In  Art.  97  we  derived  a  rule  for  the  product  of  two  bino- 
mials of  the  form  x-{-a^  a;  -}-  6,  by  considering  the  following 
cases  in  multiplication : 

1.  (ic-f  5)(.^'4-3)  =  a^  +  8a;-^15. 

2.  (x-r:>){x-B)  =  x^-8x-\'l5. 

3.  (a;  +  5)(x-3)  =  a^  +  2a;-15. 

4.  {x-6)(x-i-S)  =  x^-2x-U. 

In  certain  cases  it  is  possible  to  reverse  the  operation,  and 
resolve  a  trinomial  of  the  form  af  -\-ax-{-b  into  the  product 
of  two  binomial  factors. 

The  first  term  of  each  factor  will  obviously  be  x ;  and  to 
obtain  the  second  terms,  we  take  the  converse  of  the  rule  of 
Art.  97  : 

Pi7id  two  numbers  whose  algebraic  sum  is  the  coefficient  of  x, 
and  whose  product  is  the  last  term. 

Thus,  let  it  be  required  to  factor  a^  —  5 a;  —  24. 

The  coeflScient  of  a;  is  —  5,  and  the  last  term  is  —  24  ;  we 
are  then  to  find  two  numbers  whose  algebraic  sum  is  —  5, 
and  product  —  24.  By  inspection  we  determine  that  the 
numbers  are  —8  and  3.     Hence, 

a^-5x-24  =  (x-8)(x-i-3). 


58  ALGEBRA. 

113.  The  work  of  finding  the  numbers  may  be  abridged 
by  the  following  considerations  : 

1.  When  the  last  term  of  the  product  is  +,  as  in  Exs.  1 
and  2,  the  coefficient  of  x  is  the  sum  of  the  numbers  ;  both 
numbers  being  +  when  the  second  term  is  +,  and  —  when 
the  second  term  is  — . 

2.  When  the  last  term  of  the  product  is  — ,  as  in  Exs.  3 
and  4,  the  coefficient  of  x  is  the  difference  of  the  numbers 
(disregarding  signs);  the  greater  number  having  the  same 
sign  as  the  second  term,  and  the  smaller  number  the  opposite 
sign. 

We  may  embody  these  observations  in  two  rules,  which 
will  be  found  more  convenient  than  the  rule  of  Art.  112  in 
the  solution  of  examples  : 

1.  If  the  last  term  is  -f,  Jlnd  two  numbers  whose  sum  is  the 
coefficient  of  x,  and  whose  product  is  the  last  term  ;  and  give  to 
loth  numbers  the  sign  of  the  second  term. 

II.  If  the  last  term  is  —^find  two  numbers  whose  difference 
is  the  coefficient  of  ^^  and  whose  product  is  the  last  term;  give 
to  the  greater  number  the  sign  of  the  second  term^  and  to  the 
smaller  number  the  opposite  sign. 

ITote.  By  the  expressions  "  coefficient  of  x  "  and  "  last  term,"  in  the 
above  rules,  we  understand  their  absolute  values,  without  regard  to  sign. 

EXAMPLES. 

.     114.   1.  Factor  a^+14a;  + 45. 

According  to  Rule  I.,  we  find  two  numbers  whose  sum  is 
14,  and  product  45.  The  numbers  are  9  and  5  ;  and  as  the 
second  term  is  +,  both  numbers  are  -|-.     Hence, 

a^+14i»  +  45  =  (a;-h9)(a;  +  5),  Ans. 

2.  Factor  ic^  — 6 a; +  5. 

By  Rule  I.,  we  find  two  numbers  whose  sum  is  6,  and 


FACTORlKa.  59 

product  5.     The  numbers  are  5  and  1  ;    and  as  the  second 
term  is  — ,  both  numbers  are  — .     Hence, 

x^—6x-^6={x  —  o){x  —  l),  Ans, 

3.  FsLctorx^-j-5x-U. 

By  Rule  II.,  we  find  two  numbers  whose  difference  is  5, 
and  product  14.  The  numbers  are  7  and  2 ;  and  as  the 
second  term  is  -f ,  the  greater  number  is  -h ,  and  the  smaller 
number  — .     Hence, 

x'^6x-U  =  (x+7){x-2),  Ans. 

4.  Factor  a^  —  5x  —  24. 

By  Rule  II.,  we  find  two  numbers  whose  difference  is  5, 
and  product  24.  The  numbers  are  8  and  3  ;  and  as  the 
second  term  is  — ,  the  greater  number  is  — ,  and  the  smaller 
number  +.     Hence, 

a^-5x-24  =  (a;-8)(a;  +  3),  Ans, 

Factor  the  following : 

5.  x'  +  bx  +  G.  17.  x'-ex-U. 

6.  x'-3x+2.  18.  m^  +  Um  +  eS, 

7.  2/2 +  22/ -8.       '  19.  a^-loa  +  U. 

8.  m2-7m-30.  20.  /  +  72/-60. 

9.  a^-lla+lS.  21.  a;--lla;+10. 

10.  x^  +  x-6.  22.  m^  +  27n-80, 

11.  c2  +  9c  +  8.  23.  w2-f-23n+102. 

12.  2/' -22/ -35.  24.  a^-9a;-90.' 

13.  a2+13a-48.  25.  a- -11  a -26. 

14.  xr-10x  +  2\.                '         26.  x^-\-x-4:2. 
16.  ar^+13a7  +  36.  27.  c2-18c4-32. 
16.  n2-n-90.  28.  m2-8m-33. 


60  ALGEBRA. 

/ 

29.  a^  +  20x-{-75.  37.  o^^- 19.^'2- 120. 

30.  3^  +  405-96.  38.  c«  +  12c3  +  ll. 

31.  /-17?/-110.  39.  a^y^  +  2xf-120. 

32.  a^-19x+78.  40.  a254_  7^52_  ^44^ 

33.  x^'+lx-dS.    '^■^.  41.  nV  +  25 7ia;  + 100. 

34.  a2  +  22a  +  105.  42.  2/^-20/ +  91. 

35.  a^-23i»  +  130.  43.  o.:^b^ -2a'b^-4.8. 

36.  a^  +  10a2-144.  44.  m*  +  26m2-87. 

45.  Factor  ir^  +  5  abaP  -  84  a^^^. 

"We  find  two  numbers  whose  difference  is  5,  and  product 
84.  The  numbers  are  12  and  7 ;  and,  by  the  rule,  the 
greater  is  +,  and  the  smaller  —  c     Hence, 

a-*  +  5  abx"  -  84  a'b^  =  (x^  +  1 2  ab)  {p^-1  ab) ,  Ans, 

46.  Factor  1  —  6  a  —  27 o^. 

The  numbers  whose  difference  is  6,  and  product  27,  are  9 
and  3.     Hence, 

l-6a-27a2=  (i_9a)(l4-3a),  Ans. 

Factor  the  following : 

47.  a'-^ax  +  2x'.  66.  (a +  6)^  + 5 (a +  6)'+ 4. 

48.  ix?-\-bxy-e>Q,y\  67.  l-9a  +  8a2. 

49.  l  +  13a  +  42a2.  68.  b^  +  ^ aW  -  b2 a\      \ 

60.  m^— 15m?i  +  56w^  .69.  (m  — 71)^+ (m  — 71)  —  2. 

61.  o?-ab-b(Sb\  60.  x^ -bx^ -b()x^. 

62.  a^b^  +  4:abG-4:5c\  61.  a^  +  8ab  +  12b\ 

63.  l-3a;-10a^.  62.  1 -ISxy +  4:0a^yK 

64.  a^  +  15a3  +  44a2.  63.  (a- 6)2- 3(a- 6) -4. 

65.  ;s2_iOcc2/''2-39ajy.  64.  a^y +  80:22,22;  _  43 ;22^ 


J'ACTORING.  61 

115.  If  a  quantity  can  be  resolved  into  three  equal  fac- 
tors, it  is  said  to  be  a  perfect  cube,  and  one  of  the  equal 
factors  is  called  its  cube  root. 

Thus,  since  27  aW  equals  Sa^b  x  Sa-b  x  3a^6,  it  is  a  per- 
fect cube,  and  3a^6  is  its  cube  root. 

116.  It  is  evident  from  the  above  that  the  cube  root  of  a 

monomial  may  be  found  by  extracting  the  cube  root  of  the 
coefficient  and  dividing  the  exponent  of  each  letter  by  3. 
Thus,  the  cube  root  of  126  scf^y^x^  is  bx^y^z. 


Case  VI. 

117.  }Vhen  an  expression  is  the  sum  or  difference  of  two 
perfect  cubes. 

By  Art.  98,  the  sum  or  difference  of  two  perfect  cubes  is 
divisible  by  the  sum  or  difference  of  their  cube  roots ;  and 
in  either  case,  the  quotient  may  be  written  by  inspection  by 
aid  of  the  rules  of  Art.  98. 


EXAMPLES. 

1.  Factor  a^  +  l. 

The  cube  root  of  a^  is  a.  and  of  1  is  1  ;  hence,  one  factor 
is  a  +  1 . 

Dividing  the  expression  by  a  + 1 ,  we  have  the  quotient 
a^  -  a  +  1  (Art.  98) .     Hence, 

a3+l=(a  +  l)(a'-a  +  l),  Ans. 

2.  Factor  21:x?-Q4.f. 

The  cube  root  of  21  x^  is  3ic,  and  of  64^/^  is  4?/;  hence, 
one  factor  is  3a;  — 41/.  By  Art.  98,  the  other  factor  is 
^a?+l2xy  +  Uy''.     Hence, 

21o^-Q>^f  =  {^x-Ay){^x^-\-l2xy'{'lQy'-),  Ans, 


62  ALGEBRA. 

Factor  the  following : 

3.  a^  +  a?.  8.    a^ +  h\  13.   m^-Un\ 

4.  m^-n^.  9.    x^  +  \.  14.    64ar^-125. 

6.    a^-1.  10.    27a^-l.  15.    125a'^  + 27m». 

6.  a^y'  +  i?.  11.    8c«-d^  16.    64c3(f  +  27. 

7.  l-Sic^.  12.    27  +  8a3.  17.    Uh-B,a^b\ 

Case   VII. 

118.  TFi^ew  an  expression  is  the  sum  or  difference  of  two 
equal  odd  powers  of  two  quantities. 

By  Art,  100,  the  sum  or  difference  of  two  equal  odd  powers 
is  divisible  by  the  sum  or  difference  of  the  quantities ;  and 
in  either  case,  the  quotient  may  be  written  by  inspection  by 
aid  of  the  laws  of  Art.  99. 

EXAMPLES. 
1.   Factor  a**  4- &^ 

By  Art.  100,  one  factor  is  a-\-h.  Dividing  the  expression 
by  a  +  6,  the  quotient  is  a^  -  a^b-\-a"b'^  -  ab^  +  b^  (Art.  99). 
Hence, 

a''  +  6^  =  (a  +  b)  {a^  -  a%  +  a'W  -  ab^  +  b') ,  Ans, 

Factor  the  following : 

2.  a^  —  b'.                 5.    mJ  +  n\  8.  c^-m^nK 

3.  ar'  +  l.                   6.    x^ -y\  ^           9.  l+32n^ 

4.  l-a\                  7.    a'-l.  10.  243^;'^-/. 

11.    a;^4-128.  12.    32 -243  a''. 

119.  By  applying  one  or  more  of  the  rules  already  given, 
an  expression  may  often  be  separated  into  more  than  two 
factors. 


FACTORING.  68 

1 .  Factor  2  ao^y^  —  8  axy'^. 

By  Case  I. ,  2  aarV  —  8  ax]^  =  2  axy^'i^  —  4y^) . 
Factoring  the  quantity  in  the  parenthesis  by  Case  IV., 
%a7?f  —  %ax^  =  2axy^{x^'ly){x-1y),  Ans, 

2.  Factor  m®  —  n^. 

By  Case  IV. ,  m®  —  w*  =  (m"  +  n^)  (m^  —  n') . 

By  Case  VI. ,  m*  +  n^  =  (m  +  n)  (m^  —  mn  +  n^) , 
and  m^  —  w^  =  (m  —  n)  (m^  +  mw  +  ?i^) . 

Hence, 
m^— n®=  (m+n)  (m— n)  (m*— ?nn4-n^)  (m^-hwin+w*),  -47i«. 

3.  Factor  oi?  —  ]/^. 
By  Case  IV., 

^^f^{x'^J^y'){x'^-^-) 

=  (a;*  +  2/0  (^  +  2/0  (^'  +  y)  (^  -  2/) »  ^w«- 

MISCELLANEOUS   EXAMPLES. 

120.  In  factoring  the  following  expressions,  the  common 
monomial  factors  should  be  first  removed,  as  shown  in 
Example  1  of  the  preceding  article. 

1.  ^a^^-^o!^x.  8.    5a3-5. 

2.  i_4a;  +  4ar^.  9.    cv^U-^dK 

3.  x«-l.  10.    x'-\^. 

5.  a;2 _|_  ti.^  _|_  2>^.  _^  ci^,.  12.    3a^  +  27a'  +  42. 

6.  m--7m-8.  13.    ar^- (2^/ -  S;^)^. 

7.  2ar^  +  a;.  14.    r'= -i- eOa6  +  lOOd^. 


64  ALGEBRA. 

15.  5 a'bc- 10 ab^'c- Id abc\  21.    l  +  12a5  +  27fi;^ 

16.  3a^-21a^-\-30a\  22.    18a^''y-2xy\ 

17.  a^  +  8f^.  23.   x'-x". 

18.  2a^-2a.  24.    4ar22/*4- 28a;,v-+ 49. 

19.  l-a2-62  +  2«6.  25.    a^  +  6a-^-40. 

20.  a^-8a;+7.  26.    a^- 18«Z;-4062. 

27.  2  ar''?/  +  2  xf  -  2  a;^/^^  +  4  icy . 

28.  12m3n  -  18mV  +  24 ^mi^ 

29.  32a^5  +  4a6^  40.    (x^ +  y^ -z^ -4:x'y'. 

30.  a;^  — 81.  _    41.    a"bc  -  ac^d  -  ab^d -\- bcd\ 

31.  2/-2/^  42.    a2-14a5  +  3362. 

32.  0^3  + 2^2 -a; -2.  43.    3a;V  +  3a;/. 

33.  x^-{-7x^-S0x\  44.   477i^- 20m2/i  +  25n2. 

34.  (Sx-^yy-(x-2yy.      45.    3  a-^6  +  3  a^fe^  _  ^  ^^3^ 

35.  mV-8maj3-65.  46.    a'x^ -a-y- -b^x^ +by. 

36.  1350^ -5ar^  47.    (a  -  26)2-2(a-26)-8. 

37.  2a^y-2xY-60xy\        48.    lOOa^?/*- 81^'. 

38.  80a^f-5x^y.  49.    a«-64. 

39.  3a^b-{-18a'b-\-27ab.        50.    i«^-(a;-6)^ 

51.  (a2  +  3a)2-14(a2  4-3a)+40. 

52.  (4w  +  w)2-(2m-3n)2. 

53.  (a2-62_c2)2_462c2.       57.    a''-\-b^-c'-d^~2ab-2cd. 

54.  1000  +  27m«.  58.    (ar^  +  4)2- 16a^. 

55.  x^ —  X" —  x-{-l.  59.    a^ —  y^ —  Sxy(x  —  y). 

56.  3 (a2- 6^) -(a -6)2.         60.    (a^  +  a-^/-^. 


HIGHEST  COMMON   FACTOR.  66 


IX.  HIGHEST  COMMON  FACTOR. 

121.  A  Common  Factor  of  two  or  more  quantities  is  a 
quantity  which  will  divide  each  of  them  without  a  remainder. 

Thus,  2an/^  is  a  common  factor  of  12ary  and  20x^y^. 

122.  A  prime  quantity  is  one  which  cannot  be  divided, 
without  a  remainder,  by  any  integral  quantity  except  itself 
or  unity. 

For  example,  a,  6,  and  a-\-c  are  prime  quantities. 

123.  Two  quantities  are  said  to  be  prime  to  each  other 
when  they  have  no  common  factor  except  unity. 

Thus,  2a  and  36*  are  prime  to  each  other. 

124.  The  Highest  Common  Factor  of  two  or  more  quanti- 
ties is  the  product  of  all  the  prime  factors  common  to  those 
quantities. 

It  is  evident  from  this  definition  that  the  highest  common 
factor  of  two  or  more  quantities  is  the  expression  of  highest 
degree  (Art.  33)  which  will  divide  each  of  them  without  a 
remainder. 

Thus,  the  highest  common  factor  of  a^if  and  x^y*^  is  a?y^, 

125.  In  determining  the  highest  common  factor  of  alge- 
braic quantities,  it  is  convenient  to  distinguish  three  cases. 

Case  I. 

126.  When  the  quantities  are  monomials. 

1.  Find  the  H.C.F.  of  ^2a^b\  70  a'bc,  and  98a*b^cP. 

4.2a'b'  =  2'3-7'a^b'' 

70a-bc=2-6'7'a'bc 

98a*b^d'  =  2'7'7'a'b^d^ 


Hence,  the  H.C.F.  =  2.7- a'b{Axt.  124)  =  Ua'b,  Ans, 


ee  ALGEBRA. 

RULE. 

To  the  highest  common  factor  of  the  coefficients^  annex  the 
common  letters^  giving  to  each  the  lowest  exponent  with  which 
it  occurs  in  any  of  the  given  quantities, 

EXAMPLES. 
Find  the  highest  common  factors  of  the  following : 

2.  a^ix^,  7a*x,  5.  18  mn\  46mhi,  72  m^n\ 

3.  16cd\  dcH.  6.  lUxfz^  Uia^y:^. 

4.  bAa^b,  90ac2.  7.  ISa^o;,  45ay,  60aV. 

8.  lOSa^yV,  lUxfz*,  120a^2/V. 

9.  d6a'b\  120  a^b',  168  a%^ 

10.  SlaWii,  85 aSn^x,  llda*mY. 

Case  II. 

127.  WTien  the  quantities  are  polynomials  which  can  be 
readily  factored  by  inspection, 

EXAMPLES. 

1.  Find  the  H.C.F.  of 

5a^y-16x^ysiiid  10xPy  +  40oi^y  —  210xy. 
By  the  methods  of  Chapter  VIII. , 

5a^2/  —  16afy=:6x^y{x  —  S) 
10ix^y  +  40x^y  —  210xy=10xy{x^-^4:X~21) 
=  10xy(x  +  7)(x-S). 

In  this  case  the  common  factors  are  5,  a?,  2/,  and  a?  — 3. 
Hence,  the  H.C.F.  =  5xy(x  —  S),  Ans. 

2.  Find  the  H.C.F.  of 

4a52_4a;  +  l,  4a^-l,  and  2ax-a-2bx  +  b. 


HIGHEST  COMMON  FACTOR.  67 

4ic2-l=(2a;  +  l)(2a;-l) 
2ax-a-2bx  +  b=  (a  -  6)  (2  a;  -  1) 
Hence,  the  H.C.F.  =  2a;  —  1,  Ans, 

Find  the  highest  common  factors  of  the  following : 

3.  3  aoi^  —  2  a^x  and  aV  —  3 abx, 

4.  x^  —  y^  and  a^  +  y*. 

5.  9a^-462and(3a2-26)«. 

6.  2a^-2ar' and  6a^- 6a;. 

7.  3ca;  +  21c-3da;-21cf  andar^  — 3a7-70. 

8.  m^w  +  2mV  +  mn''  and  m*n  +  m7i*. 

9.  3ar'' +  Oa.-^- 120a;  and  Saar'-Oaa;- 30a. 

10.  3a;y-4?/  +  3a!2;-42;  and  Qa.-^— 16. 

11.  a;2_a;-42,  ar'- 4a; -60,  and  ar^+ 12a;  4- 36. 

12.  a2-l,  a«+l,  anda2  +  2a4.1. 

13.  4ar^-12a;  +  9,  4ar^-9,  and4m2wa;-6m2w. 

14.  a;^  — a;,  a.'^  +  ^a;^  — 10  a;,  and  a."^  — a;. 

16.  a3-86»,  a2-a5-262,  Siud  a^ - 4 ab  +  UK 

16.  2ar»+2a;2_4^^  3a;^+6a;»-9a;2^  and  4ar^-20a;*-f-16a;*. 

17.  8m»-125,  4m2-25,  and  4m«- 20m4-25. 

18.  a;*-16,  a.*2-a;-6,  and  (a;2_4^2^ 

19.  3aa;«-3aar',  aa;^  -  9  aa;^  ^  g  ^^.^  and  2  aar^  —  2  aa;. 

20.  a^ -h\ab-b^  +  (m- be,  and  a^ - a^b  +  ab"" - b\ 

21.  12aa!-3a  +  8ca;-2c,  16ar^-l,  and  160;^ -8a;-f  1, 


68  ALGEBRA, 


Case  III. 


128.  WJien  the  quantities  are  polynomials  which  cannot  be 
readily  factored  by  inspection. 

The  rule  in  Arithmetic  for  the  H.C.F.  of  two  numbers,  is 

Divide  the  greater  number  by  the  less;  if  there  is  a  re- 
mainder, divide  the  divisor  by  it;  and  so  on;  continuing  the 
operation  until  there  is  no  remainder.  Then  the  last  divisor 
is  the  highest  common  factor  required. 

For  example,  required  the  H.C.F.  of  169  and  546. 

169)546(3 
507 
39)169(4 
156 
13)39(3 
39 

Therefore  13  is  the  H.C.F.  required. 

129.  We  will  now  prove  that  a  similar  rule  holds  for  the 
H.C.F.  of  two  algebraic  quantities. 

Let  A  and  B  be  two  expressions,  the  degree  of  A  being 
not  lower  than  that  of  B.  Suppose  that  B  is  contained  in 
A  p  times  with  a  remainder  C ;  that  G  is  contained  m  B  q 
times  with  a  remainder  D ;  and  that  D  is  contained  in  C  r 
times  with  no  remainder.  To  prove  that  D  is  the  H.C.F.  of 
A  and  B. 

The  operation  of  division  is  shown  as  follows : 

B)  A  {p 
pB 

~C)  B(q 
q_C 

D)C(r 
rD 

We  will  first  prove  that  D  is  a  common  factor  of  A  and  B. 


HIGHEST  COMMON^  FACTOR.  69 

From  the  nature  of  subtraction,  the  minuend  is  equal  to 
the  sum  of  the  subtrahend  and  remainder  (Art.  59). 

Hence,  A=pB  +  C  (1) 

B  =  qC+D  (2) 

Substituting  the  value  of  G  in  (2) ,  we  have 

B  =  qrD-\-D  =  D{qr-^l)  (3) 

Substituting  the  values  of  B  and  (7  in  (1),  we  have 

A  ==pD  {qr-\-l)  +  rD  =  D  (pqr  -i-p  +  r)  (4) 

From  (3)  and  (4)  we  see  tliat  D  is  a  counnon  factor  of  A 
and  B. 

We  will  next  prove  that  every  common  factor  of  A  and  B 
is  a  factor  of  D. 

Let  K  be  any  common  factor  of  A  and  jB,   such  that 

A  =  mA",  and  B  =  nK.     From  the  operation  of  division,  we 

see  that 

C  =  A-pB  (5) 

D  =  B-qG  (6) 

Substituting  the  values  of  A  and  B  in  (5) ,  we  have 

C  =  mK—pnK, 

Substituting  the  values  of  B  and  C  in  (6),  we  have 

D  =  7iK—  q  {mK—piiK)  =  K{n  —  qm  +pqn) . 

Hence  K  is  a  factor  of  D. 

Therefore,  since  every  common  factor  of  A  and  J5  is  a 
factor  of  Z),  and  since  D  is  itself  a  common  factor  of  A 
and  B^  it  follows  that  D  is  the  highest  common  factor  of  A 
and  B. 

130.  Hence,  to  find  the  H.C.F.  of  two  algebraic  expres- 
sions, A  and  B,  of  which  the  degree  of  A  is  not  lower  than 
that  of  B, 


70  ALGEBRA. 

Divide  Ahy  B;  if  there  is  a  remainder,  divide  the  divisor 
by  it;  and  continue  thus  to  make  the  remainder  the  divisor, 
and  the  preceding  divisor  the  dividend,  until  there  is  no  re- 
mainder. Then  the  last  divisor  is  the  highest  common  factor 
required. 

Note  1.  Each  division  should  be  continued  until  the  remainder 
is  of  a  lower  degree  than  the  divisor. 

Note  2.  It  is  important  to  keep  the  work  in  the  same  order  of 
powers  of  some  common  letter,  as  in  ordinary  division. 

1.   Find  the  H.C.F.  of 

18ic3_51a!24-13rK  +  5  and  6i»2_i3aj-5. 

18a^-39aj2-15a7 


-12a^+28a;+  5 

~12ar+26a;+10 

2x-  5)6a^- 

-IZx- 

-5(3. 

^+1 

6a^- 

-Ibx 

2x- 

-5 

2x- 

-5 

Hence,  2fc  —  5  is  the  H.C.F.  required. 

Note  3.  Either  of  the  given  expressions  may  be  divided  by  any 
quantity  which  is  not  a  factor  of  the  other,  as  such  a  quantity  can 
evidently  form  no  part  of  the  highest  common  factor.  Similarly,  any 
remainder  may  be  divided  by  a  quantity  which  is  not  a  common  factor 
of  the  given  expressions. 

2.   Find  the  H.C.F.  of 

6a^-26a:^  +  14:XSind  6ax^ +nax—10a. 

Dividing  the  first  expression  by  x,  and  the  second  by  a, 
we  have 

ex'-25x-{-U)6af+nx-10{l 
6a^-26x-j-U 
36a?- 24 


HIGHEST   COMMON  FACTOR.  71 


Dividing  the  remainder  by  12, 

3x -  2)60^  -  26x  +  U(2x  -  7 
6a^-    Ax 

-21x-\-U 
-21X+U 

Hence,  3a;  — 2  is  the  H.C.E.  required. 

Note  4.   If  the  first  term  of  a  remainder  is  negative,  the  sign  of 
each  term  may  be  changed. 

3.    Find  the  H.C.F.  of  2o^ --Sx-2  and  2x^ ~  5x -S, 

2ar'_3a;-2)2ar^-5a;-3(l 
2a;^-3a?-2 
-2aj-l 

Changing  the  sign  of  each  term  of  this  remainder, 

2a;+l)2ic2-3a;-2(a;-2 
2aP+    X 
-4a;-2 
-4x-2 


Hence,  2a;  +  1  is  the  H.C.F.  required. 

Note  5.  If  the  first  terra  of  the  dividend  or  of  any  remainder  is 
not  divisible  by  the  first  term  of  the  divisor,  it  may  be  made  so  by 
multiplying  the  dividend  or  remainder  by  any  quantity  which  is  not  a 
factor  of  the  divisor. 

4.   Find  the  H.C.F.  of 

2a^-7a^  +  6x-6  and  Sx^ -  7x^ -7x-{-3. 

Since  3  a;^  is  not  divisible  by  2  a^,  we  multiply  the  second 
quantity  by  2. 

2a^-7x^+6x-Q)6x^-Ux^-Ux-^    6(3 
6a;^-21a;^  +  15a;-18 
7ay'-2Qx+2i 


72  ALGEBRA. 

Since  2a^  is  not  divisible  by  7a^,  we  multiply  each  term  of 
the  new  dividend  by  7. 

7a^-29a;  +  24)14a^-49a^+35a;-42(2a; 
14a;^-58a^  +  48a; 

dx^-13x-4:2 

Multiplying  this  by  7  to  make  its  first  term  divisible  by 

7a^-29a;+24)63a^-    91a;-294(9 
63x^-261  a? +216 
170a;-510 
Dividing  by  170, 

x-S)7x^-2dx-\-24:(7x-^S 
lx^-2lx 

-  8a; +  24 

-  8a; +  24 


Hence,  a;—  3  is  the  H.C.F.  required. 

Note  6.  If  the  given  quantities  have  a  common  factor  which  can 
be  seen  by  inspection,  remove  it,  and  find  the  H.C.F.  of  the  resulting 
expressions.  This  result,  multiplied  by  the  common  factor,  will  give 
the  H.C.F.  of  the  given  quantities. 

6.   Find  the  H.C.F.  of 

6  a;^  —  aa;^  —  5  a^a;  and  21  x^  —  26  aa;^  +  5  a?x. 

Removing  the  common  factor  a;,  we  find  the  H.C.F.  of 
6  a;^  —  aa;  —  5  a^  and  21  a;^  —  26  ax  +  5a^  Multiplying  the  lat- 
ter by  2, 

6ar^-aa;-5a2)42a;2-52aa;+ 10^2(7 
42a;^-    7aa;-35a^ 
—  45  aa;  +  45  o? 
Dividing  by  —  45  a, 

x  —  a)Qo?—    ax~bo?{Qx-\-ba 

^0?  —  Q>ax 

5  aa;  ~  5  a^ 
5  aa;  —  5  a^ 


HIGHEST   COMMON   FACTOR.  73 

Multiplying   x  —  a  by   a,    the   common   factor,   we   have 
«{x  —  a)  ox  x^  —  ax  as  the  H.C.F.  of  the  given  expressions. 


EXAMPLES. 
13L   Find  the  highest  common  factors  of  the  following  : 

1.  a^  +  a;  — G  and  2a;'  — llit'  +  14. 

2.  6ar^-7a;-24and  12a;2_^g^,_15 

3.  2a2-5a  +  3  and4a3-2a2-9a  +  7. 

4.  'l^x'-\-\\ax-  28a2  and  403:^ _ 5^^ q^. _^ i4^j^ 

5.  8a=^-22a2  +  5aand6a26-23a64-206. 

6.  a^  —  5  mx^  +  4  rri^x  and  7^  —  mit?'  +  3  m-x^  —  3  m^x. 

7.  5mV  +  58m?i2  4.337i2and  10m3  4-31m2- 20m  -  21. 

8.  2a*  4-  3a3a;  -  9 aV  and  Ga^  -IJa^a;  -f  Uaar^  -  3a^. 

9.  a.'^-S  anda^-6a^4-lla;-6. 

10.  2a.'»-3a^-a;  +  l  and6a;3-a^  +  3a;-2. 

11.  8m--22m7i+5n2  and  6 m*- 29771^71 +43 mV- 20 mn^ 

12.  aa:^  +  2aa^ 4-  aaJ  +  2a  and  33.-^  -V2x^  -  3a^ -  6a;. 

13.  aa;*— aa;^— 2  aa:^+ 2 aa;  and  aa;'— 3 ax'* +2  aa^^+aa;^— aa;. 

14.  2a;*-2a;»  +  4ar^  +  2a;  +  6  and  3a;*  +  6a:3_33,_g^ 

16.    a*4-ci^-6a2  +  a  +  3  and  a*  + 2a3- Ga^- a  + 2. 

16.  :x^  -  x"^  -  6a^  +  20?  -^-Q^x  B.Tidi  x^  -{-  x^  -  y?  -23?  -2x. 

17.  15a-V-20a-ar^-65a-a;-30a2 

and  126a:3  +  206a.-2_166a;-166. 

18.  a^-\-a?x-\-a'x^  +  a3?-\x'^ 

and  a^  -!-  2  a^x  +  8  a V  4-  4  aa;^  _  j  q  aj*. 


74  ALGEBRA. 


19.  x*-^x^-\-x^-l  &ndx^-\-3x^-\-2x. 

20.  x*-a^y-3a^y^-]-5xf-6y''      , 

and  Sx'^  —  5x^y  —  x^y^  —  7xy^-{-  lOy*. 

21.  2x'^—5a^-\-5x^-5x+3  and  2x^-7 x^-\-4.:f^ -{-5x-?j. 

22.  3a^-2a^6  +  2a262_5a63-26* 

and  6a^-  a^b  +  2  a^S^  -2ab^-  h\ 


132.  To  find  the  H.C.F.  of  three  or  more  quantities,  find 
the  H.C.F.  of  two  of  them ;  then  of  this  result  and  the  third 
quantity,  and  so  on.  The  last  divisor  will  be  the  H.C.F.  of 
the  given  quantities. 

EXAMPLES. 
Find  the  highest  common  factors  of  the  following : 

1.  2x^-bx  —  ^2,  4aj2  +  8ic-21,  and  6a^4-23a.'  +  7. 

2.  12a?2_28a;-5,  14  a^- 39  a; +10,  and  10a^-lla;-35. 

3.  6m2+7mn  +  2w2,  ^m^ —  Im^n- 12 mn'^ -  4. n"^, 

and  15m^-f  4mn  — 4w^. 

4.  6a2+13a-5,  6a3+19a2+ 8a- 5, 

and  3a3  +  2a2  +  2a-l. 

5.  a^  +  3a^-6a;-8,  i»^  +  5a^  +  2a;-8, 

andar^-3a^-16a;  +  48. 

6.  a^-7a;  +  6,  a;^  +  3a^-16a;  +  12, 

anda^-5a^4-7a;-3. 

7.  2a»-3a2-5a  +  6,  2a^  +  3a2_8a-12, 

and2a3-a2-12a-9. 


LOWEST   COMMON  MULTIPLE.  75 


X.    LOWEST    COMMON    MULTIPLE. 

133.  A  Common  Multiple  of  two  or  more  quantities  is  a 
quantity  which  can  be  divided  by  each  of  them  without  a 
remainder. 

Hence,  a  common  multiple  of  two  or  more  quantities  must 
contain  all  the  prime  factors  of  each  of  the  quantities. 

134.  The  Lowest  Common  Multiple  of  two  or  more 
quantities  is  the  product  of  their  different  prime  factors, 
each  being  taken  the  greatest  number  of  times  which  it 
occurs  in  any  one  of  the  quantities. 

It  is  evident  from  this  definition  that  the  lowest  common 
multiple  of  two  or  more  quantities  is  the  expression  of 
lowest  degree  which  can  be  divided  by  each  of  them  without 
a  remainder. 

Thus,  the  lowest  common  multiple  of  oi^y'^  y^z,  and  a^2^  is 

When  quantities  are  prime  to  each  other,  their  product  is 
their  lowest  common  multiple. 

135.  In  determining  the  lowest  common  multiple  of  alge- 
braic quantities,  we  may  distinguish  three  cases. 

Case  I. 

136.  When  the  quantities  are  monomials. 

1.  Find  the  L.C.M.  of  ma^x,  60ay,  and  8^ca;». 
SGa^o;  =2'2-^'^'a''x 
60ay=2-2'3-6'ay 
84ca^  =2.2.3.7.caj3 


Hence,  the  L.C.M.  =  2.2.3.3.5.7. a^ca^f  (Art.  134) 
=  1260  a^cxhf,  A71S. 


76  ALGEBRA. 

RULE. 

To  the  lowest  common  multiple  of  the  coefficie7its,  annex  all 
the  letters  which  occur  in  the  given  quantities^  giving  to  each 
the  highest  exponent  which  it  has  in  any  of  the  quantities. 

EXAMPLES. 
Find  the  lowest  common  multiples  of  the  following : 

2.  Qa^b,  a^b\  6.  a'b%  da^b\  12a^b\ 

3.  10x^y,12y^z.  7.   16x^y,  4:2y^z. 

4.  30m\  21.n\  8.  8c^d^  lOac,  ISa^d. 

5.  Q>ab,  106c,  14ca.  9.  24m3a^,  SOw^y,  ^2xyK 

10.  2>^xyh\  63arV,  28x^2^. 

11.  ^Ott^bd?,  90  ac^d^  bWcd^. 

Case  II. 

137.  When  the  quamtities  are  p)olynoniials  which  can  be 
readily  factored  by  inspection. 

1.  Find  the  L.C.M.  oi  x^+x-Q>,x^  -  ^x-\- 4.,  and  a:^- 9a;. 
x^j^x-Q,=^    (a;+3)  (a;-2) 

a;S_9a;  =  a;(fl;+3)  (a;-3) 

Hence,  the  L.C.M.  =  x{x  -  2)\x  +  3)  (a;  -  3) ,  (Art.  134) 
=  a;(a;-2)2(a,'2-9),  Ans. 

EXAMPLES. 
Find  the  lowest  common  multiples  of  the  following : 

2.  y?  —  y^  and  xy  —  y^. 

3.  xr  —  1  and  ar  — 7a;  — 8. 


LOWEST   COMMON  MULTIPLE.  77 

4.  Sa'b  +  8ah'  and  6a -6b. 

5.  m^  —  n^  and  m^  —  n^. 

6.  a  —  b  and  cr  —  iab  +  Sb^. 

7.  ay^  —  2xy-\- 1/  and  a^y  —  xy^. 

8.  2a2  +  2a6,  3ab-Sb\  and  4:  crc- 4  b^c. 

9.  aj2  +  2aaj-35a2andiB2-2aa;-15a2. 

10.  mn  -f-  n^,  mw  —  ?i^,  and  ??i^  —  ??^. 

11.  ax-2a-^bx-2b  Sinda^-2ab-3b\ 

12.  aic^  +  a^x,  ^  —  a^,  and  x^  —  a^. 

13.  8(a--62),  G(a4-6)2,  and  12(a-6)2. 

14.  a^-10x2  +  21a:andaar^  +  5aa;-24a. 

15.  x'-l,x'-2x+l,iinda^  +  2x-^l. 

16.  2-2a^,  4-4a?,  8  +  8a;,  and  12  +  12ar'. 

17.  x2  +  5ic  +  4,  a^  +  2£c-8,  andar*+7ic  +  12. 

18.  a(a;  —  6)  (a;—  c) ,  5(a;  —  c)  (a;  —  a) ,  and  c(x  —  a){X'-  b). 

19.  (2m-l)2,  4m2-l,  and8m3-l. 

20.  a^  -^a,a^-  a^,  and  a«  +  a^. 

21.  a^  — 4a-f  3,  a2_|_«_i2,  and  a^  — a  — 20. 

/  22.   l-cc*,  l  +  2a;2^^4^  ^^jj(j  l_2a;2_^^4^ 
/ 

23.  (a  +  6)2  _  c«  and  (a  -  c)^  -  b\ 

24.  aa;  —  a?/  —  6a;  +  by,  {x  —  y) ^,  and  3  a^^  —  3  ab^. 

25.  9a^  +  12a^  +  4a;,  18aaj*-12aaj3  +  8aa^, 

and  27ar^  +  8. 

26.  x^-y'^-z'-i-2yz&ndx'-y^  +  z'  +  2^. 


78  ALGEBRA. 

Case  III. 

138.  When  the  quantities  are  polynomials  which  cannot 
be  readily  factored  by  inspection. 

Let  A  and  B  be  two  expressions  ;  let  F  be  their  highest 
common  factor,  and  M  their  lowest  common  multiple.  Sup- 
pose that  A  =  aF  and  B  —  bF;  then, 

AxB  =  abF'  (1) 

Since  a  and  b  can  have  no  common  factor,  the  L.C.M.  of 
aF  and  bF  is  abF;  that  is,  Jf=  abF;  whence, 

FxM=abF'  (2) 

From  (1)  and  (2)  we  have  AxB  =  Fx  M  (Art.  42,  7). 
That  is,  the  product  of  any  two  quaritities  is  equal  to  the 

product  of  their  highest  common  factor  and  lowest  common 

multiple. 

Hence,  to  find  the  L.C.M.  of  two  quantities. 

Divide  their  product  by  their  highest  common  factor ;  or. 

Divide  one  of  the  quantities  by  their  highest  common  factor, 

and  multiply  the  quotient  by  the  other  quantity. 

139.  1.  Find  the  L.C.M.  of 

6a^-nx-^12  andl2x^-4:X--21. 

6x^-nx-{-V2)12x^-    40^-21(2 
12a^-34a;  +  24 
30x-4:5 
2x-    S)Gx^-nx  +  12(Sx-4: 

-  8.'«+12 

—  8aj-f-12 


That  is,  the  H.C.F.  of  the  quantities  is  2a;  —  3.     Dividing 
6x^  —  nx-i-12  by  2a;  — 3,  the  quotient  is  3a;  — 4. 
Hence,  the  L.C.M.  =  (3a;-4)(12a;-- 4a;- 21) 

=  36a;3-60a;--47a;  +  84,  Ans. 


LOWEST   COMMON  MULTIPLE.  79 

EXAMPLES. 
Find  the  lowest  common  multiples  of  the  following : 

2.  2x'^  +  x-Q  and4:X^  —  8x  +  S. 

3.  6«2_f.i3^_28  and  12a;2-31a;-|-20. 

4.  8ic2_^30a.^7  and  12a;2-29aj-8. 

5.  ea^-8a^-S0xsindQax^-{-19ax+15a. 

6.  a' -8 ah  -\-lW  and  a^  -9a?h-\-  23  ab^  -  15 h\ 
^ 2m^n  —  Smn  —  2n  and  2m*  —  Qm^  +  6m^  —  87n  -j- 8. 

8.  Goic^-ci^a;- 12a3and  lOaa^  — 17a2a;  +  3a^ 

9.  a^  +  a2-8a-6  and  2a3  — 5a2-2a  +  2. 

10.  2a^  +  a^-a;  +  3  and  2ar»  +  5a^-a;-6. 

11.  «» -2a'b  +  2ab^-  b^  and  a^  +  «'&  -  aM_  63. 

12.  x*  +  2oi^-{-2a^-\-xsindax^—2ax  —  a. 

13.  2ic^-lla^  +  3a;2  4-10a;and  3.t^- Ua:^  _  (3^^_5a,^ 
14.-a;*-a;»-8a;  +  8  and  x' -8x^ -^9x -2. 

140.  To  find  the  L.C.M.  of  three  or  more  quantities,  find 
the  L.C.M.  of  two  of  them  ;  then  of  this  result  and  the 
third  quantity  ;  and  so  on. 

EXAMPLES. 
Find  the  lowest  common  multiples  of  the  following : 
(l.    a^— 1,  2ic2_9a;4-7,  and  2ar'  +  3a;  — 5.* 
<2.    3a2_2a-l,  6a2- a- 1,  and  9 a^- 3a -2. 
V3.    2a:2_5^^2,  4ar^4-4a;-3,  and  lOar^- 7a;  + 1. 
4.    ix--6x-18,  4:X^-i-Aa^-3x,  and  6x* -}- oa^ -6a^, 
6.    a^-Ga^H-lla-e,  a«-cr-14a  +  24, 
and  a^  +  a^-Ha +15. 


80  ALGEBRA. 


XI.    FRACTIONS. 


141.    The  expression  -  signifies  a-i-b;  in  other  words,  - 

b  b 

denotes  that  a  units  are  divided  into  b  equal  parts,  and  that 

one  part  is  taken. 

Or,   what  is  the  same  thing,  -  denotes  that  one  unit  is 

b 

divided  into  b  equal  parts,  and  that  a  parts  are  taken. 


142.  The  expression  -  is  called  a  Fraction ;  a  is  called  the 

b 

numerator,  and  b  the  denominator. 

By  Art.  141,  the  denominator  shows  into  how  many  parts 
the  unit  is  divided,  and  the  numerator  shows  how  man}^  parts 
are  taken. 

The  numerator  and  denominator  are  called  the  terms  of  the 
fraction. 

143.  An  Entire  Quantity  or  Integer  is  one  which  has  no 
fractional  part ;  as  2xy,  ov  a-\-b. 

Every  integer  may  be  considered  as  a  fraction  whose  de- 
nominator is  unity  ;  thus,  a  =  — 

144.  A  Mixed  Quantity  is  one  having  both  entire  and 

fractional  parts  ;  as  a  +  -,  or  a;  H 

%  2  y  +  z 


GENERAL  PRINCIPLES. 

145.  If  the  nnmerator  of  a  fraction  be  multiplied,  or  the 
denominator  divided,  by  any  quantity,  the  fraction  is  multi- 
plied by  that  quantity. 


FRACTIONS.  81 


I.  Let  -  be  any  fraction.     Multiplying  its  numerator  by 

etc  CLG  a 

c,  we  have  —     To  prove  that  —  is  c  times  — 
h  b  b 

In  each  of  these  fractions  the  unit  is  divided  into  b  equal 
parts ;  in  the  first  case  ac  parts  are  taken,  and  in  the  second 
case  a  parts.     Since  c  times  as  many  parts  are  taken  in 

—  as  in  -,  it  follows  that 

II.  Let  —  be  any  fraction.     Dividing  its  denominator  by 

oc 

c,  we  have  —     To  prove  that  -  is  c  times  —  • 
b  b  be 

In  each  of  these  fractions  a  parts  are  taken  ;  but  since  in 
the  first  case  the  unit  is  divided  into  b  equal  parts,  and  in  the 
second  case  into  be  equal  parts,  the  parts  in  -  will  be  c  times 

as  great  as  in  ».     Hence,  ' 

be 

146.  If  the  numerator  of  a  fraction  be  divided,  or  the  de- 
nominator multiplied,  by  any  quantity,  the  fraction  is  divided 
by  that  quantity. 

I.    Let  —  be  any  fraction.     Dividing  its  numerator  by  c, 

we  have  -•     To  prove  that  ^  is  —  divided  by  c. 
0  b       b 

By  Art.  145,  (1),  cx^  =  ^. 

0      b 

Whence  it  follows  that        ^  =  ^  -^.  c. 

b      b 


82  ALGEBRA. 


II.    Let  -  be  any  fraction.     Multiplying  its  denominator 
h 

by  c,  we  have  —     To  prove  that  —  is  -  diyided  by  c. 
be  be      b 

By  Art.  145,  (2),         cx^  =  ^- 

be      b 

Whence  it  follows  that       —  =  -  -^  c. 

be      b 

147.  If  the  numerator  and  denominator  of  a  fraetion  be 
both  multiplied,  or  both  divided,  by  the  same  quantity,  the 
value  of  the  fraction  is  not  altered. 

For,  by  Arts.  145  and  146,  multiplying  the  numerator  mul- 
tiplies the  fraction,  and  multiplying  the  denominator  divides 
it.  Hence,  the  fraction  is  both  multiplied  and  divided  by 
the  same  quantity,  and  its  value  is  not  altered. 

Similarly  we  may  show  that  if  both  terms  are  divided  by 
the  same  quantity,  the  value  of  the  fraction  is  not  altered. 


TO  REDUCE  A  FRACTION  TO  ITS  LOWEST  TERMS. 

148.  A  fraction  is  in  its  lowest  terms  when  its  numerator 
and  denominator  are  prime  to  each  other. 

Case  I. 

149.  When  the  numerator  and  denominator  can  be  readily 
factored  by  inspection. 

Since  dividing  both  numerator  and  denominator  by  the 
same  quantity,  or  canceling  equal  factors  in  each,  does  not 
alter  the  value  of  the  fraction  (Art.  147),  we  have  the  fol- 
lowing rule : 

Resolve  both  numerator  and  denominator  into  their  prime 
factors,  and  cancel  all  which  are  common  to  both. 


FRACTIONS.  83 

1.  Reduce — r-  to  its  lowest  terms. 

45  (rltx 

18  a^6^c  ^  2  ♦  3  •  3  » a%-c 
Aba'b^x      S'S'd-a'b^x 

Dividing  both  terms  by  3  •  3  •  a^b^,  we  have  — -,  Ans. 

0  X 

rj^   27 

2.  Reduce  — to  its  lowest  terms. 

ic2-2a;-3 

x'  —  'll     _  (a;  -  3)  (a^  +  3a;  +  9)  _ar^  4- 3a; +  9   ^^^^ 
ar2_2a;-3"~        (a;-3)(a;  +  l)  ^+1       ' 

Note.  If  all  the  factors  of  the  numerator  be  removed  by  cancella- 
tion, unity  (which  is  a  factor  of  all  algebraic  expressions)  remains  to 
form  a  numerator. 

If  all  the  factors  of  the  denominator  be  removed,  the  result  is  an 
entire  quantity;  this  being  a  case  of  exact  division. 


EXAMPLES. 

g     a^y^J  g  32?7i7t  g     15mxy^ 

xif^  '  56m*w^  Ibmx^y^ 

^     2a'b'c  -  ebx'f:^  -^     115c^a;^y 

ba%(^'  '  2Qx'fz'  '      2Z&^' 

5     12  V  3  54  aV  ^^     154mV 

'     32ar^*  *  12(h^bc  '    8Sm^xf' 

^2     2a^cd-^2abccl  ^q     6a^b-j-3a^b\ 

Ga'xy  +  Qabxy  '    3a-b^-^Qab^' 

^g       3ar^-6a;^y  ^^     4c^-20c  +  25_ 

6x'y'-12xf'  '        4cS-25c 

14      ^a^y-Q^y ,  13     m^- 10m +  16 

•   aj2_g3,_|.i5*  •      m2  +  m-72  * 

j^g      g^  — 2  a  — 15  ,  ^^g  9an^  — 4  a 


a2+i0a  +  21  96712-126^1  +  46 


84  ALGEBRA. 


20.     .^^-/^'    „.  25. 


21.    ^i£_iir ..  26. 


a2  4-a6-662 

Sa^-j-f 

4:a:^-2x'7j  +  xy' 

ac  —  ad  —  bc-^bd 

a^-W 

aa?  —  Aa 

a?-^:x?J^Ux 

27?/^- 125 

x^- 

-a^-\-2x- 

-2 

2x' 

+  x^  +  Ax 

+  2 

x'- 

-4:X-{-16 

ax' 

^  +  64  ace 

a?- 

-(b-^cY 

23  ga^  —  4 g  oo         (a.-^— 4)  (a;^— 3a;+2) 
*"""■*  '  (a^-4aj+4)(a.'2+x-2) 

24  27y^-125  ^9  (a-by-jc-dy 
92/' -30?/ +  25  *  (a-c)2-(6-(Z)2 


Case  II. 

150.  When  the  numerator  and  denominator  cannot  be 
readily  factored  by  inspection. 

Since  the  H.C.F.  of  two  quantities  is  the  product  of  their 
common  prime  factors,  we  have  the  following  rule  : 

Divide  both  numerator  and  denominator  by  their  highest 
common  factor. 

EXAMPLES. 

1.  Reduce  — -^ ^^^  to  its  lowest  terms. 

6a2_a-12 

By  the  rule  of  Art.  130,  the  H.C.F.  of  2a^-5a  +  3  and 
6a'  —  a  —  12  is  2a  —  3.  Dividing  the  numerator  by  2 a  —  3, 
the  quotient  is  a  —  1  ;  and  dividing  the  denominator,  the 
quotient  is  3  a  +  4.     Hence, 

2g^-5a  +  3^   a-1      ^^^ 
6a'-a-12      3a  +  4' 


FRACTIONS.  85 

Reduce  the  following  to  their  lowest  terms  : 

x^^Gx-\-5  .-      a^  +  af-Sx-2 

8. 

2a^-\-5x^-2x-{-3 

g      6y^-19/  +  7y  +  12_ 
6f-26y'-\-ny  +  20 

2a^-3a'-a-2 


4. 


3x'  +  4.x-7 

lOa^-a-21 

2a'-7a  +  6 

2m2-5m-f-3 

12wi2-28m  +  15 

x'-2x-S 

af-2x'-2x-S 

12m2  +  16mn  — 3n* 

jj      a^-4ar^y  +  4a^^-y^  ^  / 


10m2  +  m?i-2l7i^  ""'    a^-2oi^y-{-4:X7f-3f" 


151.  Since  a  fraction  represents  the  quotient  of  its 
numerator  divided  by  its  denominator,  it  is  positive  when 
its  terms  have  the  same  sign,  and  negative  when  they  have 
different  signs. 

Thus,  if?=^a;, 

0 

then  =  X,  and  — —  =  — -  =  —  x. 

—  6  0—0 


152.   It  follows  from  Art.  151  that  the  fraction  -  can 

h 


written  in  any  one  of  the  forms 


a        —  a  a 

6'       ~h~'  ^^       ^' 


That  is,  if  the  signs  of  both  numerator  and  denominator 
are  changed^  the  value  of  the  fraction  is  not  altered.  But  if 
the  sign  of  either  one  is  changed ^  the  sign  before  the  fraction 
is  changed. 


86  ALGEBRA. 

153.  If  either  numerator  or  denominator  is  a  polynomial, 
care  must  be  taken,  on  changing  its  sign,  to  change  the  sign 
of  eoc^  of  its  terms. 

Thus,  the  fraction      ~    ,  by  changing  the  signs  of  both 
c  —  ct 

numerator  and   denominator,   can  be  written   in  the   form 
^  '    or  (Art.  67). 


—  {c—d)         d—c 

rom  Art.  151  that  the  fraction 

cd 


154.   It  follows  from  Art.  151  that  the  fraction  —  can  be 


written  in  any  one  of  the  forms 

(-^)?>  (-^)(-b)       (-a){-b) 

c{-dy  cd         '     (_c)(-d)' 

(-a)6  ab  (-a)(-5) 

or, — ,      — — -, — — ,  eic. 

cd  (—c)d  c(—d) 

From  which  it  appears  that 

If  the  terms  of  a  fraction  are  composed  of  factors^  the 
signs  of  any  even  number  of  factors  may  be  changed  without 
altering  the  value  of  the  fraction.  But  if  the  signs  of  any 
odd  number  of  factors  a7'e  changed,  the  sign  before  the  frac- 
tion is  changed. 

Thus,  the  fraction  — can  be  written  in  any 

{x  —  y){x  —  z) 
one  of  the  forms 

a  —  b                      b  —  a                          b  —  a  . 

'     7 ^7 ^'     —- ^^ ^,  etc. 


(y  —  x){z  —  x)       {y  —  x){x  —  z)  {y  —  x){z-x) 

TO    REDUCE  A    FRACTION    TO    AN    ENTIRE    OR    MIXED 
QUANTITY. 

155.    Since  a  fraction  is  an   expression  of   division,  we 
have  the  following  rule  : 

Divide  the  numerator  by  the  denominator. 


FRACTIONS.  87 

1.  Reduce  ,  ^  ""    ^^—     ^q  g^  mixed  quantity. 
Dividing  each  term  of  the  numerator  by  the  denominator, 

3x  3x       Sx       Sx  3x 

8a^-12a^-9a;  +  10  ,  .      , 

2.  Reduce ' to  a  mixed  quantity. 

4ar  — 3  - 

4a;2-3)8a;^- 12a^-9a;+10(2ic-3 

8af -6x 

-12ar^-3ic 
-12a^  4-9 

-3x-\-l 

A  remainder  whose  first  term  will  not  contain  the  first  term 
of  the  divisor,  may  be  written  over  the  divisor  in  the  form 
of  a  fraction,  and  added  to  the  quotient.     Thus,  the  result  is 

3a; -fl 


2a;-3  + 


4a;2-3 


Or,  since  the  sign  of  each  term  of  the  numerator  may  be 
changed,  if  at  the  same  time  the  sign  before  the  fraction  is 
changed  (Art;  152),  we  have 

8a^-12x^-9x^l0^^^_^_3x-l     ^^^^ 
4a;2-3  4:0^-3 

EXAMPLES. 
Reduce  the  following  to  mixed  quantities  : 


3. 

5a^-10a;  +  4 

6.   2^-^l 

5x 

a;-3 

A 

6a?  —  3a^+dx- 
3x 

-2, 

Y     a3-a2-a-2 

a^  4-  a  -  1 

5 

a^  +  2f 

Q     Uo?-8x  +  l 

x+y 

4a;-l 

88  ALGEBRA. 

9     ^L±A\  12  aT^  +  2a^  +  3a;4-4 

'     a-\-b'  '  x'-^x-^l 

2  ??i  —  3  72  a;  +  y 

11     2a^-a^-9a^+14  ^^  6ar^-13a;^+ 6a;- 6 


TO    REDUCE  A    MIXED    QUANTITY  TO    A    FRACTIONAL 

FORM. 

156.  The  operation  being  the  converse  of  that  of  Art.  155, 
we  have  the  following  rule  : 

Multiply  the  integral  part  by  the  denominator ;  odd  the 
numerator  to  the  product  when  the  sign  before  the  fraction  is 
+ ,  a7id  subtract  it  when  the  sign  is  —  ;  and  write  the  result 
over  the  denominator. 

1.  Reduce  — ^^^ f-  ^  —  2  to  a  fractional  form. 

2a;-3 

By  the  rule, 

a^-5    .^      ^^a;-5+(a^-2)(2a;-3) 
2a;-3  2a;-3 

^a;-5  +  2a^-7a;  +  6 
2a;-3 

^2^-6^+1    ^^^^^ 
2a;-3 

(j,2 7^2  K 

2.  Reduce  a-\-b to  a  fractional  form. 

a  —  b 

a^-b'-  5  _  (a  +  b)(a-  b)-(a'~b'-5) 

a-\-  0 —  —   — »  —^ 

a—b  a—b 

a—b  a—b 

Note.  If  the  numerator  is  a  polynomial,  it  will  be  found  convenient 
to  enclose  it  in  a  parenthesis,  Avhen  the  sign  before  the  fraction  is  — . 


FRACTIONS.  89 


EXAMPLES. 
Reduce  the  following  to  fractional  forms : 

3.  0^+1 +?±i.  11.   ^±1'-1. 

4.  a;  +  l — .  12.   m-?i  +  — ^ — 

5. 1-  m  —  w.       13.   a^  —  ab-\-b^  — 


6.  7x_3-5i^^ll2.  14.   ^_3x-Ml=^. 

8  a;  — 2 

7.  l_!!Lz:^.  As.         m»  +  »^    ,-(m-«). 

m  +  71  f          m'*  +  mn  +  71'' 

8.  a  +  6-^-±^.  16.    l-f-2.T4-4ar^  +  -^^tl. 

a  +  6  2a;-l 


9. 


2 


H-3a;-2.  17.    x^2y-   ^    "^-^f 

2«  +  l  iB2_4iC2/4-42/' 


10.    ^2_^2_^a&(ct  +  ?>).        18.    a^-2^'+3-'^+^^^-^ 
a  — 6  ic^4-3.^'— 2 


TO  REDUCE  FRACTIONS    TO    THEIR    LOWEST    COMMOl^ 
DENOMINATOR. 

157.  1.  Reduce  —V?  -^^  and  — ^  to  equivalent  frac- 
tions  having  the  lowest  common  denominator. 

The  lowest  common  denominator  is  the  lowest  common 
multiple  of  3a^6,  2a6^,  and  4a^6,  which  is  \2a^lr. 

By  Art.  147,  both  terms  of  a  fraction  may  be  multiplied 
by  the  same  quantity  without  altering  its  value.     Hence, 


^0  ALGEBRA; 

Multiplying  both  terms  of  -^  by  4a6,  we  have  ?^£^ 


Multiplying  both  terms  of  — -  by  6  a^,  we  have — -• 

^^    ^  2ab^    ^  12a^b^ 

Multiplying  both  terms  of  — j-  by  36,    we  have f-^* 

Therefore  the  required  fractions  are 

20  abed     18  a^mx         -.    dbny      a 

inr-,    :ri    and  ^,  Aj[is. 

\2a^b^'     12  a'b''  12  a'b'' 


It  will  be  observed  that. the  terms  of  each  fraction  are 
multiplied  by  a  quantity  which  is  obtained  by  dividing  the 
lowest  common  denominator  by  its  own  denominator.  Hence 
the  following  rule : 

Find  the  lowest  common  multiple  of  the  given  denominators. 
Divide  this  by  each  denominator  separately,  multiply  the  cor- 
responding numerators  by  the  quotients,  and  write  the  results 
over  the  common  denominator. 

Note.  Before  applying  the  rule,  each  fraction  should  be  in  its 
lowest  terms. 

EXAMPLES. 

Reduce  the  following  to  equivalent  fractions  having  the 
lowest  common  denominator  ; 

2     Sab     2ac        ^  6bc 
.   ,   ,  and 

14'     21'  6 

3.  —     —    and  — • 
a^a^    aa?^  a?x 

.     4c-l        ,  36-2 

4.    —  and — 

8a62  12  ah 


6a^y    8fz    ^^       lOxz' 


FRACTIONS.  91 


2^         and     ^^ 


a^  +  a  -  6  a'  -  4 

7.   -A-  and       ^ 


iB2-l  a;«-l 


8.   m,    -,    and  — r — 


9.   -^,   -5-,  and       ^ 


r  10       ^y  ^^         and        ^^ 

a6  ^^^      m-n 


am  —  6m  +  aw  —  6/1  2  a^  —  2  a6 

^  +  ^      ,    ^^+L_,   and  ^^  +  ^      . 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS. 

158.    It  follows  from  the  definition  of  Art.  141  that 

a     h     a-\-h         ,  a     h     a  —  b 

_-!--=:  — I — ,  and = . 

c      c  c  c      c  c 

Hence  the  following 

RULE. 

To  add  fractions,  reduce  them,  if  necessary,  to  equivalent 
fractions  having  the  lowest  common  denominator.  Add  the 
numerators  of  the  resulting  fractions,  and  write  the  sum  over 
the  common  denominator. 

To  subtract  one  fraction  from  another,  reduce  them  to 
equivalent  fractions  having  the  lowest  common  denominator. 
Subtract  the  numerator  of  the  subtrahend  from  that  of  the 
minuend,  and  write  the  result  over  the  common  denominator. 

Note.  The  final  result  should  be  reduced  to  its  simplest  form. 


92  ALGiifeM. 

1.    Required  the  sum  of and — 

The  lowest  common  denominator  is  12  ab'^c.  Multiplying 
the  terms  of  the  first  fraction  by  3  6^,  and  of  the  second  by 
2  a,  we  have 

4a-l      S-5b^^l2ab^-Sb^     6a-10ah^ 
4ac  66^0  Uab'c  12ab'c 

_12ab^-Sb^-^6a-  lOab^ 
~  12  ab^c 


^2ab^-Sb^  +  ea^^^^ 
12  ab'c 

2.    Subtract  ^^-1  from  ^^-2. 
2x                   3a 

The  L.C.D.  is  6 ax.     Hence, 

6a  — 2      4a;— 1      12ax  —  4:X 

12ax-3a 

3a            2x              6ax 

6  ax 

12ax-4:X- 

-  {12ax-3a) 

6 

ax 

12ax  —  4:X  — 

■12ax-[-3a 

6  ax 

_3a-4:X      . 

IS. 

Qax 

Note.  If  a  fraction,  whose  numerator  is  a  polynomial,  is  preceded 
by  a  —  sign,  care  must  be  taken  to  change  the  sign  of  each  term  in  the 
numerator  before  combining  it  with  the  others.  It  is  convenient  in 
Buch  a  case  to  enclose  the  numerator  in  a  parenthesis,  as  shown  in 
Ex.2. 

EXAMPLES. 
Simplify  the  following : 
3.    2x-5  J  3a;  +  ll  4,       ^  ■• 


12  18  5a62      2a^h 


f^RACtiONS.  93 


g     2a  +  3      3a  +  5  y     6  — 4a     a  +  56 

6  8      *  *      24a  30&   ' 


6     ^  —  2      2  — 3mn^  g     a— 6      2a+&      3a— 6 

2mn  3mV  4  6  8 


9     q'  +  l      6a'  +  l      &-2 
So"  12a3  66  * 


10     2a;-l     2rc  +  3     6a;+l 
12     "^15  20    ' 


-|     m  +  2      mH-2      m  +  3 
7  14  21 


12     2      2a;-l      3x^  +  1 
*    3         Qx  day"    ' 


13     a;  — 2      3a;+l      6a;  — 5      3 


14    3a  +  l      26-1      4c-l      6d  +  l, 
12a  86  16c  24d 


15.    Simplify -J— +      ^ 


a;  +  ^*^     a?  —  ar^ 

The  L.C.M.  of  a;  +  a^  anda;  — a;^  is  a;(l -|-a;)  (1  —  a;),  or 
a;(l— a;^).  Multiplying  the  terms  of  the  first  fraction  by 
1  —  a;,  and  of  the  second  by  1  +  a?,  we  have 

1  1      ^     1-a;  1+x 

x-\-QC^     x—af     a;(l— a.*^)      a;(l  — a^) 

1  — x  +  l+aJ 


a;(l  —X') 
a;(l  —  ar) 


94  ALGEBRA. 


16.    Simplify  «  +  *      «-*        *«^ 


a  —  h      a-\-h      a^  —  h^ 
The  L.C.D.  is  o?  -  b\     Hence, 

a-^b     a  —  h       Aab 
a  —  b      a-\-b      a^  —  b^ 

^{a-hby      (a -by       Aab 

a2_62  ^2_J2  ^2_52 

^{a-\-by-(a-by-4:ab 
a'-b' 

^  g^  +  2  a6  +  &^  -  (a^  -2ab +  b^)-4:ab 
o?-b' 

Simplify  the  following : 


17. 

l+x     l-x 

24. 

{m  —  ny  '  m^  —  r? 

18. 

26. 

1                     1 

a2_4a  +  4     a^  +  a-G 

19. 

1             1 

26. 

X          Zx          2xy 
x  —  y     x-\-y     a?  —  y^ 

20. 

a            b 
a  —  b     a-f-6 

27. 

a            b           2ab 
a-\-b     a-b      a^-b^ 

21. 

a+6     a—b 
a—  b     a-\-b 

28. 

m              11 

mn  —  n^     m  —  n     n 

22. 

x+y     2xy  +  x'^ 

29. 

x+y     x-y      ^ 
x—y     x+y 

y      y{^  +  y) 

23. 

1-^x     1  —  x 
1—x      1 +» 

30. 

2          3           2a;-3 

X      2x-\      4a^-l 

FRACTIONS.  96 

Si   -1-     I      1            2a          go       1  X            3?-^x 

'a-^l'^a-h      a'  +  W           '  l-x  {l-xY      {l-xf 

I  1  2cd 


33. 


ah  —  cd     ab-{-  cd     -a^b^  —  (?d? 


34     fl;-3  x-\-\  a;  +  13 

x-'l  x-\-h      ^-\-'6x-\^ 

«g     x  —  a  X—  h {a  —  hy 

x—h  x  —  a      {x  —  a){x  —  b) 


1  1         .       ^ 


x{x  +  l)      x{x-l)      ar^-1 

37  «4-^         .  ^-f-c         .         c  +  g 

(6-c)  (c-a)      (c-a)  (a-b)      {a-b)  (6-c) 

38.   -i ^-f-^. 

og         2x-6 x+2 a?  +  l 

40.        ^-y       I        y-^ g-a; 

(a;+2;)  (2/+0)       (aj+y)  (a;+2;)       (a;+y)  (.7+2;) 

In  certain  cases,  the  principles  of  Arts.  152  and  154  en- 
able us  to  change  the  form  of  a  fraction  to  one  which  is  more 
convenient  for  the  purposes  of  addition  and  subtraction. 

41.    Simplify  ^+^i:_. 

a  —  b      b^  —  or 

Changing  the  signs  of  the  terms  in  the  denominator  of  the 
second  fraction,  and  at  the  same  time  changing  the  sign 
before  the  fraction  (Art.  152),  we  have 

_3 25  +  a 

a-b      a'-b^' 


96  ALGEBRA, 

The  L.C.D.  is  now  a^  —  6^.     Hence 

3         2b-j-a_3{a  +  b)      2b  +  a 
a-b      o?-b''~    d'-b^        a^-W 


3(a  +  &)-(26  +  o^) 
a?-b^ 

3a  +  36-2&-a_2a  +  5      . 


42.    Simplify 
1 


{x-y){x-z)      (y-x){y-z)      {z-x){z-y) 

By  Art.  154,  we  may  change  the  sign  of  the  factor  y  —  x 
in  the  second  denominator,  at  the  same  time  changing  the 
sign  before  the  fraction ;  and  we  may  change  the  signs  of 
both  factors  of  the  third  denominator.  The  expression  then 
becomes 

I + 1 ?_. 

(x-y){x-z)       {x-y){y-z)       {x-z){y-z) 

The  L.C.D.  is  now  {x  —  y){x  —  z)(y  —  z).  Hence  the 
result 

=  (y-g)4-(a?-g)  — (a;-2/)  ^  y -z-{-x-z-x-{-y 
{x-y){x-z){y-z)  {x-y){x-z){y -z) 

2y-2z  ^  2(y-g) 

(x-y){x-z)(y-z)      (x-y)(x-z){y -z) 

2 


(x-y){X'-z) 


,  Ans. 


Simplify  the  following : 

43  _JL_+_2 45    _l_-f_L_. 

*"•    a'-ab^b'-ab  '    Sx-x'^x'-Q 

44  5a  +  l      3a-l  ^  1 1__ 

'    3a  — 3      2  — 2a  '    m^  —  mn     n^  —  m? 


FRACTIONS. 

47. 

(a-2){x-\-2)      {2-a){x-^a) 

48. 

a             a            2oL- 

a  +  5      h-a      a?-W 

49. 

X               X              x^ 
l+x      l-x     x^-1 

50. 

2-x     x-^      «2_5a._,.g 

51. 

^               I              ^ 

1 

(a-6)(5-c)       {h-a){a-c)       (c- 

-a)(c- 

-6) 

52. 

2                              3 

1 

97 


(a;-2)(a;-3)       (3-a;)(4-a;)      (a;-4)(2-a;) 


MULTIPLICATION    OF    FRACTIONS. 


ft  /• 

159.   Required  the  product  of  -  and  — 

b  d 

2  5  5 

In  Arithmetic,  -  times  -  signifies  two-thirds  of  -• 

Similarly,  in  Algebra,  ^  X  -  signifies  a  bths  of  -•     That 
0     cl  d 

is,  we  divide  -by  b,  and  multiply  the  result  by  a, 
d 

By  Art.  146,11.,  |-&  =  ^- 

a  oa 

By  Art.  145,1.,  ^xa  =  ^. 

bd  bd 

TT  a  ^c      ac 

Hence,  I  X  3  =  13* 

b     d     bd 


98""  ALGEBRA. 

"We  have  therefore  the  following  rule  for  the  multiplication 
of  fractions : 

Multiply  the  numerators  together  for  the  numerator  of  the 
product,  and  the  denominators  for  its  denominator. 

Mixed  quantities  should  be  reduced  to  a  fractional  form 
before  applying  the  rule. 

Common  factors  in  the  numerators  and  denominators 
should  be  canceled  before  performing  the  multiplication. 

EXAMPLES. 

36V 


L.    Multiply  1^  by  i^ 
^^    96aj2      ^  4.a 


lOa^      3  6V  ^  10 . 3 .  a^6Vy 
96a^       4  ay       d-A-a^xy'' 

5b^x 


6y 


Ans. 


2.   Multiply  together  3^-^^,  ^,  and  ^^  +  ^    ■ 
ar  —  2a;  —  3    ar  —  x  x^-{-x  —  6 


X^  —2X  sy   ^  —^    sy  X^+  X 

X  ~~z X 


^       a;(a;-2)  (a;  +  3)(a;-3)  a;^a;+l) 

(a;-3)(a;H-l)  a;(a;-l)  (a;  +  3)(a;-2) 

-,  Ans. 


x-1 

Multiply  the  following : 

o      ba^hc       -,0  t2a6c        -,56 

3.   -andSmn.  6.   — , — ,  and  — 

Umn"  36   5a  d>c 

4.  ?^and^.  6.    «4l^%ndi4. 
5a/          ^ahy?  9y^   le;^^  10 a;^ 


FRACTIONS.  99 


7.   3«^,  3:^,  and  §^.         12.    ^^2a&+^  ^^^       6 


4cc?'  26(i'  96c  '  a  +  b  ax—hx 

9.   ^A^  and  -11-       14.   i^!^^!±f  and  4^- 
10.   ^^16  and  ^=25 .         15.   1  ,  4  _  5    „^        3aj 


a^4-5x  a^  — 4a;  x     o?         x^-j-x  —  2 

11.   _«^iL  and  4r^^     16.   i^^  i^,  and  -J- 

(i^  +  2 a6  a^  —  ah  \  —y    x  +  or  \—x 

jg^       a;^y-4y      ^^^^  ^-^-^. 
{x-yf-z^  xyr\-2y 

19.  ^-y"     ^      ^  +  2/"     ^andl  |      ^ 

20.  ^^-(^-^)^  and  ^^-(^  +  < 
^a  +  cY-h'  {a-cy-b' 


DIVISION    OF   FRACTIONS. 

160.   Required  the  quotient  of  -  divided  by  -- 

b  d 

By  Art.  85,  we  are  to  find  a  quantity  which,  when  multi- 

plied  by  -,  will  produce  -• 
d  b 

That  quantity  is  evidently  ^  ;  hence, 

be 

a  .  c  __ad 
b     d  ^hc 


TOO  ALGEBRA. 

We  observe  that  the  quotient  is  obtained  by  multiplying 

the  dividend,  -,  by  -,   which  is  the  divisor  inverted.     We 
h         c 

have  then  the  following  rule  for  the  division  of  fractions  : 
Invert  the  divisor^  and  proceed  as  in  multiplication. 

Mixed  quantities  should  be  reduced  to  a  fractional  form 
6efore  applying  the  rule. 

If  the  divisor  is  an  integer,  it  may  be  written  in  a  frac- 
tional form,  as  explained  in  Art.  143. 


EXAMPLES. 

^'   ^"^^^5^^^' 10^- 

By  the  rule, 

6a^b   .    9a^b^  ^  ea^b      lOar^y^^  Ay     ^^^^ 
5a^2/'  *  lOitY      5a.y      9a^b^      Wx 

2.   Divide  — ;r-  by  — ^^-— 

lo  5 

a^-9  .  fl^  +  2a;-3^  (a;  +  3)(a;-3)  5 

15      *  5  15  (a;  +  3)(a;-l) 

^-^  .,Ans 


3(aj-l) 


Divide  the  following : 

6.  .rrJ-T^by  1 


o?j^a-12    -"  a2-f-3a-18 
4:      x"    ^  12^3 


FRACTIONS.  101 


w      ar^  —  25 a;  ,        x^  —  hx 

8.   "t-^:„by  ^ 


o^  +  ^ab  +  y    '  G?-IP- 


m^  — 2m7i  +  7i^         m  —  n 


10.  4+f  by  !^=^. 

11.  9+-^by3  +  -A2L. 

j2         a^-8a&^      ^    a^  +  2a^6  +  4a6» 
a2-2a6-362    ^  a-36 


18.    A_A_f_Aby-? ^. 

32/'     xy      23?    ^  3f     It? 


COMPLEX  FRACTIONS. 

161.  A  Complex  Fraction  is  one  having  a  fraction  in  its 
numerator  or  denominator,  or  both. 

It  may  be  regarded  as  a  case  in  division  ;  its  numerator 
answering  to  the  dividend,  and  its  denominator  to  the  divisor. 

EXAMPLES. 
1     Reduce  ^^  its  simplest  form. 


,_c      bd  —  c  bd  —  c     bd  —  c 

"d        d 


102  ALGEBRA. 

It  is  often  advantageous  to  simplify  a  complex  fraction  by 
multiplying  both  numerator  and  denominator  by  the  lowest 
common  multiple  of  their  denominators. 


2.   Reduce  t — ^ — 2Jl^  to  its  simplest  form. 
b      .      a 

a—b     a+b 

The  L.C.M.  of  a  +  5  and  a  —  6  is  (a  +  b)(a  —  b).  Multi- 
plying each  of  the  component  fractions  by  (a  -\-b)(a  —  b)t 
we  have 

a(a-\-b)  —  a(a  —  b)_a^-\-ab  —  a^  +  ab_    2ab       j 
b{a-\-b)-ha{a-b)  ~  ab -j- b' +  a' -  ab  ~  a'  +  b'' 


3.    Reduce  to  its  simplest  form. 

X 


1  =^+1  ^+\Ans. 


X 


Reduce  the  following  to  their  simplest  forms : 

or—^  2,1  a^  +  4v^       .   , 

^ x^  -\--  — ! — - —  4  a; 

4.   ^.  *  6.    -.  8.    l. 

l+i  1+- 

X  « 

1      1  1 

T-~  a-2+- 

K     0      a  m  a 

0.     -.  7.    . 

a__6  1  _i 

6     a  a 


1 

y 

2 

'  X 

9. 

x-l^ 

12 

■  a; 

a;  +  3- 

18 

X 

FRACTIONS. 

17. 

m      m  —  n 

n^        mn 

1+  '^ 


1-x' 


1  2     I         ^^ 

1  —X^-\- 


1-ar^ 


.         12  a4-  6  ■  g  — 6 

aj  +  3  c  — d     c  H-d 


12.   2 L^.  19.  ^ 


^  3  +  -^  aj4         ^ 


^-2  1  I  ^  +  1 

2  3-a; 


a_6^  a;4-2y     x 

13.  -i-^.  20       ^+^     ^ 

a      J      6  *    a;4-22/         a; 


b  a  y  x-\-y 


14.   t—-^.  21.   -4±-' 

y       X  a-^x 


1  1  a?^h^      a?-h^ 


103 


j^     l-x     \+x  a'-b'     a^-j-b^ 

1       .      1  *      a+^      g— 6 

1 —X     1 +«  a—  b     a  +  b 


^      26  — 2c  m  —  ?i      m^  +  y^^ 

g  +  6  —  c  m  +  71     m^  —  71^ 


16.   4^ 23. 


j^  ■        2  c  m^         m^n  +  yi^ 

g--6  — c  m  —  n      (m  —  ny 


104  ALGEBRA. 

MISCELLANEOUS    EXAMPLES. 
162.    Eeduce  the  following  to  their  simplest  forms : 
J  26      a  +  bx-j-ca^  g     10 a^ -{- SO ab +  20 b^ 

3m«-75m  V  ^A  V 

g     a^(l  +  xY-a^(l  +  xy  g       l-ax  +  a(x+a) 

{l+xy  '  *    (l-axy+(x  +  ay 


A       a^  —  b^    ,    a^  —  ab^ 
{m-\-ny  '  bm  -\-bn 


'  (i-^D-e-;) 


f>     1  +2a^       2  +  a;  ^q     b(b  —  ax)  +  a(a  +  &a;) 

2  4-2a^      2  + 2a;*  *     {b  -  axy  +  {a  +  bxy 

J  J         aa;  b       .  ax(Sb  —  ax) 

ax  +  b  ax~b         (^y?  —  b^                   _^ 


j2    _Jrx-^^_^     10a;  +  10a^ 


13. 


2  +  4a;+2a^      9-18a;  +  9a;* 
6n^-487i^ 

971^ +  1871^^  +  36923* 


14.   a:2_2^2_6^1ii^Vf^. 
^  x-Sy 


^5.    ^  +  &      <^ 


462 


a  —  b     a-\-b     a?  — 


''■  (^+^+^^)(^-3- 


--  2a;  — 1 2a;  +  l 

2a;2_2a;4.i      2a;2^2a;  +  l' 


FRACTIONS.  105 

1 


18. 

+ 


14-^      2\a;  —  a      x-\-aJ 


a' 

X 


1Q       ^  —  y     ^  +  y  20      ^  —  9^  + 26  a;  —  24 

""^^ ~'  '    ar^- 123.-2  + 47a; -60 

ai       2      o    J.      oj"      62(7a  +  6&) 

21.  a^  —  Sab  — 2b- ^^ —■ — ^• 

a  — 3b 

22.  M±l_l y  a^' 

a;  +  2/  2/  — a^      a^  — i/* 

no     (a?  +  y  +  ;g)^  +  (a;-y)2+(y-g)'  +  (g-a;)^ 

24        1  4  8(1 -aO 

'    a;-2      (.T-2)2      (a;-2)» 

«-      a*  —  a*b  —  ab^  +  6^ 


a*  —  a^&  —  a-62  +  ab^ 

26       (4a:  +  .v)2-(a:-2i/)^ 
•    (^Sx-4yy-(2x  +  Syy 

27.    _1_  +  J_  +  _1 (a  +  6  +  c)^ 

a  +  6      6  +  c      c  +  a      (a  +  6)  (6  +  c)  (c  +  a) 

28         ^(1-^^-) 1-2^  I       2 

(14-aj)(l  +  9a;)       (l.+  a;)  (1 +4a;)      l+4a; 

gg        1  1       .     3a;2  3a^ 


x-1      x+1      a^+1      a:3_i 
OQ     fct-\-b      a^  +  6^   .   fa  —  b      a^  —  b 


b     a'-by      \a-\-b     a^-\- 


106  ALGEBRA. 


XII.    SIMPLE   EQUATIONS. 

163.  An  Equation  is  a  statement  of  the  equality  of  two 
expressions. 

Tlie  First  Member  of  an  equation  is  the  expression  on  the 
left  of  the  sign  of  equality,  and  the  Second  Member  is  the 
expression  on  the  right  of  that  sign. 

Thus,  in  the  equation  2a;  —  3  =  3a;  —  5,  the  first  member  is 
2a;  — 3,  and  the  second  is  3a;  — 5. 

The  sides  of  an  equation  are  its  two  members. 

164.  A  Numerical  Equation  is  one  in  which  all  the  known 
quantities  are  represented  by  numbers  ;  as 

2a;  — 3  =  3a;  — 5. 

165.  A  Literal  Equation  is  one  in  which  some  or  all  the 
known  quantities  are  represented  by  letters  ;  as 

2a;  +  3a  =  6a;-4. 

166.  An  Identical  Equation  is  one  whose  two  members  are 
equal,  whatever  values  are  given  to  the  letters  involved ;  as 

a?  —  a^  —  {x-\-  a)  {x  —  d). 

167.  The  Degree  of  an  equation,  in  which  there  is  but  one 
unknown  quantity,  is  denoted  by  the  highest  power  of  the 
unknown  quantity  in  the  equation.     Thus, 

2x  —  3  =  3a; 5  ) 

„  \  are  equations  of  the  ^rs^  degree. 

and  orx  —be  —d) 

3a;^  —  2a;  =  65  is  an  equation  of  the  second  degree^  etc. 

168.  A  Simple  Equation  is  an  equation  of  the  first  degree. 


SIMPLE  EQUATIONS.  107 

169.  The  Root  of  an  equation  containing  but  one  unknown 
quantity,  is  the  value  of  the  unknown  quantity ;  or,  it  is  the 
value  which,  when  put  in  place  of  the  unknown  quantity, 
makes  the  equation  identical. 

Thus,  the  equation  5a;— 7  =  3a;-|-l,  when  4  is  put  in  place 
of  a;,  becomes  20  —  7  =  12  +  1,  which  is  identical.  Hence 
the  root  of  the  equation,  or  the  value  of  a:,  is  4. 

Note.  A  simple  equation  has  but  one  root ;  but  it  will  be  seen  here- 
after that  an  equation  may  have  two  or  more  roots. 

170.  The  solution  of  an  equation  is  the  process  of  finding 
its  roots. 

A  root  is  verified,  or  the  equation  satisfied,  when,  on  sub- 
stituting the  value  of  the  root  in  place  of  its  symbol,  the 
equation  becomes  identical. 

171.  The  operations  required  in  the  solution  of  an  equa- 
tion are  based  upon  the  following  general  principle,  which  is 
derived  from  the  axioms  of  Art.  42  : 

IftJie  same  operations  he  performed  upon  equal  quantities, 
the  results  will  he  equal. 

Hence, 

Both  members  of  an  equation  may  he  increased,  diminished, 
multiplied,  or  divided  hy  the  same  quantity,  without  destroying 
the  equality. 

TRANSPOSITION. 

172.  Any  term  may  he  transposed  from  one  side  of  an 
equation  to  the  other  hy  changing  its  sigii. 

For,  consider  the  equation  x-\-a=zh. 
Subtracting  a  from  both  members  (Art.  171),  we  have 
x-\-a  —  a=zh  —  a ; 
or,  by  Art.  26,  x  =  h  —  a. 


108  ALGEBRA. 

where  -\- a  has  been  transposed  to  the  second  member  by 
changing  its  sign. 

Again,  consider  the  equation  x  —  a  —  b. 

Adding  a  to  both  members  (Art.  171),  we  have 

x  —  a-{-a  =  b-\-a; 

or,  xz=b-^a. 

where   —  a  has  been  transposed  to  the  second  member  by 
changing  its  sign. 

Note.   If  the  same  term  appear  in  both  members  of  an  equation 
affected  with  the  same  sign,  it  may  be  suppressed. 

173.  The  signs  of  all  the  terms  of  an  equation  may  be 
changed  without  destroy  trig  the  equality. 

For,  consider  the  equation  a  —  x=b  —  c. 

Transposing  each  term  (Art.  172),  we  have 

c  —  b  =  x  —  a; 

or,  x  —  a=c  —  b, 

which  is  the  same  as  the  original  equation  with  every  sign 
changed. 

SOLUTION    OF    SIMPLE    EQUATIONS. 

174.  1 .  Solve  the  equation  5x  —  7  =  3x-{-l. 

Transposing  the  unknown  quantities  to  the  first  member, 
and  the  known  quantities  to  the  second,  we  have 

5x  —  Sx  =  7  +  U 

Uniting  the  similar  terms,     2x  =  8. 

Dividing  both  members  by  2  (Art.  171), 

aj=  4,  Ans, 


SIMPLE  EQUATIONS.  109 

Note.  The  result  may  be  verified  by  substituting  the  value  of  x  in 
the  given  equation,  as  shown  in  Art.  169. 

We  have  then  the  following  rule  for  the  solution  of  a 
simple  equation  containing  but  one  unknown  quantity  : 

Transpose  the  unknown  terms  to  the  first  member^  and  the 
knoicn  terms  to  the  second. 

Unite  the  similar  terms,  and  divide  both  members  by  the 
coefficient  of  the  unknown  quantity. 

EXAMPLES. 
2.  Solve  the  equation  14  —  5 .t  =  19  +  3a;.  , 

Transposing,        — 5a;  — 3a;=19  — 14. 

Uniting  terms,  —8x=  5. 

5 
Dividing  b}^  —8,  «  =  — -,  Ans. 

8 

Note.  To  verify  this  result,  put  x  =  — -  in  the  given  equation.   Then, 

8 


H-5(-|)=19H.3(-|) 


Or,  "  +  f  =  l«-f 

Or,  1^  =  131;  which  is  identical. 

8  8 

Solve  the  following  equations  : 

3.  8a;  =  5a;  +  42.  9.  5a;  +  14=  17 -3a;. 

4.  7a;4-5  =  -30.  10.  3a; - 31  =  11a;- 16. 

5.  7a;  +  5  =  a;  +  23.  11.  18  -  7a;  =  18a;- 7. 

6.  9a;  +  7  =  3a;-ll.  12.  27 -f  10  a;  =  13  a;  +  23. 

7.  3a;-8  =  5a;  +  8.  13.  19a;- 11  =  15  +  6a;. 

8.  5-6a;=l-4a;.  14.  32a;- 15  =  7  +  65a;. 


110  ALGEBRA. 

16.    13a;-81  =  5a;-31a;-159. 

16.  12a;-20a;4-13  =  9a;-259. 

17.  Solve  the  equation 

(2x-3y-x{x  +  l)  =  S{x-2)(x-\-  7)  -5. 
Performing  the  operations  indicated,  we  have 

4  ar*  -  1 2  a;  +  9  -  ic2  -  a;  =  3  0^  +  1 5  a;  -  4  2  -  5 . 
Transposing, 
•     4iB2__i2a;-a^-a;-3a:2-15a;  =  -42-5-9. 
Uniting  terms ,  —  28  a;  =  —  5  6 . 

Dividing  by  —  28,  a;  =  2,  Ans» 

Solve  the  following  equations  : 

18.  3  +  2(2a;4-3)=2a;-3(2a;  +  l). 

19.  2a;-(4a;-l)  =  5a;-(a;-l). 

20.  7(aj-2)-5(a;  +  3)  =  3(2aj-5)-6(4a;-l). 

21.  3(3aj4-5)-2(5a;-3)  =  13-(5a;-16). 

22.  (2a;-l)(3a;4-2)  =  (3a;-5)(2a;+20). 

23.  (5-6a;)(2a;-l)  =  (3a;  +  3)(13-4a;). 

24.  (a;-3)2-(5-a;)2  =  -4a;. 

26.  (2a;  - 1)2-  3 (a;  -  2)  +  5 (3 a;  -  2)  -  (5  -  2xy=i^  0. 

26.  2(a;-2)2-3(a;-l)2+a^  =  l. 

27.  (a;  -  1)  (a;  -  2)  (a;  +  4)  =  (a;  +  2)  (a;  +  3)  (a;  -  4) . 
.  5(7+3a;)-(2a;-3)(l-2a;)-(2a;-3)2-(5-fa;)  =  0. 
.  (5a;-l)2-(3a;  +  2)2-(4a;-3)2  +  4  =  0. 

30.    (2a;  +  l)»  +  (2a;-l)3=16a;(a^-4)-228. 


SIMPLE  EQUATIONS.  Ill 

SOLUTION  OF  EQUATIONS  CONTAINING  FRACTIONS. 

175.   1 .  Solve  the  equation = 

^  3       4       6       8 

The  L.C.M.  of  3,  4,  6,  and  8  is  24.     Multiplying  each 
term  of  the  equation  by  24,  we  have 


16a;- 

-30 

=  20a;- 

-27 

16aj- 

-20a; 

=  30    - 

-27 

-4a; 

=  3 

X 

3 

Ans. 

We  have  then  the  following  rule  for  clearing  an  equation 
of  fractions : 

Multiply  each  term  by  the  lowest  common  multiple  of  the 
denominators. 

EXAMPLES. 
Solve  the  following  equations  : 

2.  a;  +  ^  +  ^  =  -ll.  7.   ^-^-1^^^ 

2      3  '^         ""        ^'^ 

3.  ^_:^  +  i-  =  o. 

4        6       18 

4.  2a;-^  =  i^-^. 

4       14      7 

5.  7^_7  =  ^^-i^.  10.   a;-^+20  =  ^  +  ^  +  26. 

4  3         4  7  2      4 

*    6      2a;      4      12a;*  '   a;      2a;      12      3a;* 


5        20       10      4 

8. 

X      2x            2x 

9. 

X  .  11      a;_a;     3a; 
2      ~6       3~6~T' 

112  ALGEBRA. 

12.    Solve  the  equation  ^^^^  -  i^^  =  4  + 1^+^. 
^  4  5  ^     10 

Multipljing  through  by  20,  the  L.C.M.  of  4,  5,  and  10, 

15a;  -  5  -  (16 a;  -  20)  =  80  +  14a;  -f-  10 
15  a;  -  5  -  1 6  a;  +  20  =  80  +  14  a;  +  10 
15a;  -  16a; -  14a;  =  80  +  10  +  5  -  20 
—  15  a;  =  75 

a;=  —  5,  Ans. 

Note.  K  a  fraction  whose  numerator  is  a  polynomial  is  preceded  by 
a  —  sign,  care  must  be  taken  to  change  the  sign  of  each  term  of  the 
numerator  when  the  denominator  is  removed.  It  is  convenient,  in  such 
a  case^  to  enclose  the  numerator  in  a  parenthesis,  as  shown  in  the  above 
example. 

13.   3^  +  ^^^  =  ^^-  14.   x-^-^±l  =  5x-L 

7  2  5  3 

15.    7a;-li^^  =  3a;+7. 


16.   4a; -^^^+- (a; -9)  =  5a;. 
3  2^  ^ 


17.   a;-(3a;-4)-5-=^=2. 


18     ^^  =x     7  1  ^-^^.  19    a;  +  l     a;  +  4_a;-4 

'21  15   *  '2  5  7    ' 

20.    2-I^:zi  =  3x-i^^+^. 
6  4 

OI     5a;  — 2      3a;  +  4      7a;  +  2      a;— 10' 


^^-        3 

4               6               2 

22.   i(^  +  l)- 

2a;-5_lla;  +  5      a;-13 
5                10               3 

SIMPLE  EQUATIONS.  113 

3  9  2^  ^  6 

i+^^l/3^_2)_li^±2_l(2_9aj). 
7         2^  ^  14  3^  ^ 

OK     2a;H-l      4a;  +  5      8  +  3?  ,  2a;4-5 

26  ^^~^  _  7  — 3a;  _  10a;  — 3  _  3  — 5ag 

2  3a;     ~        4  2a;     * 

27  3 ^'  +  7  _  4(ar^-2)  _ a;«+16  ^  7 

2  3a;  Bx"         2 

28.  Solve  the  equation  — -^ —  =  0. 

^  a;-l      a;+l      ar^-1 

Multiplying  each  term  by  ar^  —  1,  the  L.C.M.  of  the  denom- 
inators, 

2(a;+l)  -3(a;-l)  -1=0 

2a;  +  2-3a;  +  3-l  =  0 

2a;-3a;  =  -2-3  +  l 
—  a;  =  — 4 
X  =  4,  Ans. 

oa     o  1      i.u  i.-       6a;H-l       2a;  — 4       2a;— 1 

29.  Solve  the  equation  — — = 

^  15         7a;-16  5 

Multiplying  each  term  by  15, 

^  30a;-60^g^_3^ 

7a; -16 

Transposing  and  uniting  terms,    4  =  — ^-^^ 

Multiplying  by  7a; -16,   28a;- 64  =  30a; -60 

-2a;  =  4 

a;=  —  2,  Ans. 


114  ALGEBRA. 

Note.  If  the  denominators  are  partly  monomial  and  partly  polyno- 
mial, it  is  often  advantageous  to  clear  of  fractions  at  first  partially ; 
multiplying  by  a  quantity  which  will  remove  the  monomial  denomi- 
nators. 

Solve  the  following  equations  : 

30.  — ^  =  0.  34. 

3a;-7      3a;H-7 

31.  2^nl  =  2^±I.  36. 
3a;  +  4      3a;-f2 

32.  l^zi7^±A_3.  36. 
2ar2  +  5a;-13 

33.  ^-i2_^5fl^  +  7.  37^ 
«(«  — 1)       a^— 1 

2  1 


38. 


X      x^  — 
3      ^x- 

■6x 

-7 

.2 
'3' 

{x-¥by 

^     5a 

'  +  1 

x-^ 

5 

1    1 

2 

a; 

3 

a.  +  l' 

aj  +  2 

+  3 

3a;  +  2 

2aj- 

-1 

a; 

6 

3ic- 

-7 

2 

1 

a;  —  2      a;~3      a;^  —  5a;H-6 


39     6a;  +  7      2  (a;- 1)  ^  2a;  + 1. 
15  7a;-6  5 

40.   -^ ? ^  =  0. 

1  —X      1  -fa;      1  — ar 

?a^4-3a;       1 
2a;  +  l       3a; 


41.   2^+^4-^  =  «;  +  l. 


42.    2f^-±iU3f^  =  5. 
\x  +  2)^    \x+\) 


43.        ^  1 


3a;4-l      x  +  l      2x 

44  ^  =  ^  +  1  _  7  — 2a;^_ 
9~     3  l-9x' 

45  (a;  +  i)^^a;-4 

(a;  4- 2)2      a; -2 


SIMPLE  EQUATIONS.  115 


3a^4-a;— 1       3a;  +  l' 


47    a;-l      a;  + 1  ^  2(ar^  +  4a;  +  l)^ 
a;_2"^a;4-2  (a;  +  2)2 

^     4a;4-3      12a;  — 5      2a;— 1_q 
10  5a; -29  5      ~    ' 

a;— 1      a;  —  2      a;  —  3      a;  —  4 


49. 


x  —  4:      a;  — 5 


SOLUTION  OF  LITERAL  EQUATIONS. 

3.76.   1 .    Solve  the  equation  2aa;  —  36  =  a;  +  c  —  3aa;. 
Transposing  and  uniting  terms,  5aa;  —  a;  =  36  +  c. 
Factoring  the  first  member,     a; (5a—  1)  =36  + c. 
Dividing  by  5  a  —  1 ,  x  =  — i-^ ,  Ans. 

2.    Solve  the  equation  (b  —  cxy  —  (a  —  cxy  =  6(6  —  a). 

Performing  the  operations  indicated, 

62  _  26ca;  +  c^ar^  -  (a"  -2acx  +  c2^)=  6^  -  ab 

b'-2bcx-^(^x^-a^  +2acx-c^x^  =b^-ab 

2  oca;  —  2  6ca;  =  a^  —  ab 

Factoring  both  members,     2 ca;(a  — 6)  =  a(a  —  6) 

Dividing  by  2c{a  -  6),  x=  ^(^-^) 

2c(a  — 6) 

=  ^,  Ans. 
2c 


116  ALGEBRA. 

EXAMPLES. 
Solve  the  following  equations  ; 

3.  2ax-{-d  =  3c  —  bx. 

4.  6  bmx  —  6  an  =15  am  —  2  bnx. 

5.  x  +  l  =  2ax—a^{x—l), 
o     d?  ,  h      W  .  a 

e.  — !--  = Ht* 

X  2  X  4: 

7.  {d'-2xY  =  (J.x-3a'){x  +  a^), 

8.  (2m  +  3a;)(2m-3a7)  =  n2-(3a;-n)^ 

9.  ^II^_^±^+2  =  0. 

h  a 

10.    (a;-a-&)2-(x  — a)(a;-6)  +  a5  =  0. 
a;         x  +  2h     a^  +  ft^ 


11 


a;  — a'     ic+a      x^  —  a^ 


j2         (6-3a;)(c+2a;)      ^  ^ 
2(a;-c)(6-3c-3a;) 

.    13.    (ic+a)2-(£c-a)»-a(3a;-a)(2ic+a)  =a;(a-|-l)+3. 

x+  2n 

15.  (a-a5)(6-a;) -a(&  +  l)  =  -  +  a;^ 

16.  ^_3+-^  =  A-2a(2-3a). 
2a  4a«      3a2  ^  ^ 

17    a?  I  1  —  2aa;  i  2a;—  1  _q 

^^^  2"^      2a  a^     ~    * 


f"^ 


\ 


SIMPLE  EQUATIONS.  117 


18.   JL-=l±m  =  JL-^m-l). 
mn         Sn         3n 


19.   a;4-2a     x-3a^^ 
X  —  a        ic  +  a 


20.    i^ZL^-^-±-^=l. 
2x  —  a     x  —  a 


21     E  _  <^  —  ^^^  _  ^  _  0^  —  4  5a; 
2         26c     ~  6c  3bc 

ax-i-b        36        aV  +  62 


22. 


ax  —  b     ax-\-b     a  V  —  6^ 
ax  —  b     bx  —  a  a  —  b 


f\ 


ax-\-b     bx-^a      {ax -\- b)  {bx -\- a) 


QA     x  —  n     x^  —  mx  —  '^  _  .,    , 


m  mx  —  w  mx  —  inf 

SOLUTION  OF  EQUATIONS  INVOLVING  DECIMALS. 

177.   1.  Solve  the  equation  .2ic-. 01-. 03a;  =  .113a;+. 161. 
Changing  the  decimals  into  common  fractions, 
2x        1         3x      113a;  ,    161 


10      100      100     1000      1000 

Clearing  of  fractions, 

200a;  -  10  -  30ic  =  113a;  +  161 
57a;  =  171 
a;  =  3,  Am. 


118  ALGEBRA. 

Or,  we  may  solve  the  equation  as  follows : 
Transposing,  .2a;  — .03x  — .113a;  =  .01  +  .161. 
Uniting  terms,  .057a;=.171. 

Dividing  by  .057,  a;  =  3,  Ans. 

EXAMPLES. 
Solve  the  following  equations : 

2.  .23fl;- 2.05  =  . 02a;— 1.882. 

3.  .001a;-.32=.09a;  — .2a;-.653. 

4.  .3a;-.02-.003a;  =  .7-.06a;-.006o 

5.  .3(1.2a;-5)=U+.05a;. 

>/.    6.    .7(a;+.13)  =  .03(4a;-.l)+.5. 


8.    4.25 -I?  =  11-1-=:^. 


Q     .6a;  +  .044      .5a;  — .178       oo 

W. =.d8. 

.4  .6 


f      10     2  — 3a;       5a;       2a;  — 3  _  a;  — 2      25^ 
^  *       1.0         1.25  9      ~    1.8        9° 


PROBLEIVIS.  119 


XIII.  PROBLEMS. 

LEADING    TO  SIMPLE  EQUATIONS   CONTAINING  ONE 
UNKNOWN  QUANTITY. 

178.  For  the  solution  of  a  problem  by  Algebra  no  gen- 
eral rule  can  be  given,  as  much  must  depend  on  the  skill 
and  ingenuity  of  the  student.  A  few  suggestions,  however, 
may  be  found  of  service  : 

1.  Express  the  unknown  quantity^  or  one  of  the  unknown 
quantities^  by  one  of  the  final  letters  of  the  alphabet. 

2.  From  the  given  conditions,  Jind  expressions  for  the  other 
unknown  quantities,  if  any,  in  the  problem. 

3.  Form  an  equation  in  accordance  with  the  conditions  of 
the  problem. 

4.  Solve  the  equation  thus  formed. 

PROBLEMS. 

179.  1.  What  number  is  that  to  which  if  four-sevenths 
of  itself  be  added,  the  sum  will  equal  twice  the  number 
diminished  by  27? 

Let  X  =  the  number. 

4x 

Then,  -—=  four-sevenths  of  it, 

7 

and  2x  =  twice  it. 

By  the  conditions,  x -\ =  2a:  —  27 

7x  +  ix=Ux-189 
-8a:=-189. 
Whence,  x  =  63,  the  number  required. 

2.  A  is  three  times  as  old  as  B,  and  eight  years  ago  he 
was  seven  times  as  old  as  B.  Required  their  ages  at 
present. 


120 


) 

ALGEBRA. 

Let 

X  =  B's  age. 

Then, 

Sx=  A's  age. 

Also, 

j:  —  8  :=  B's  age  8  years  ago, 

and 

Zx  —  S  =  A's  age  8  years  ago. 

By  the  conditions,  3;r  —  8=7(x  —  8) 
Sx-S  =  1x-b6 
-4t=_48. 
Whence,  x  =  12,  B's  age, 

and  3ar==36,  A's  age. 

Note.  In  the  ahove  solution  we  say  "  Let  x  =  B's  age,"  meaning 
"  Let  X  =  the  number  of  years  in  B's  age."  Abbreviations  of  this 
nature  are  often  used  in  Algebra ;  but  it  should  be  remembered  that 
they  are  in  fact  abbreviations,  and  that  x  can  only  represent  an  ab- 
stract number. 

3.  A  had  twice  as  much  money  as  B  ;  but,  after  giving  B 
$35,  he  had  only  one-third  as  much  as  B.  How  much  had 
each  at  first? 

Let  X  =  what  B  had  at  first. 

Then,  2x  =  what  A  had  at  first. 

After  giving  B  $35,  A  had  left  2  0;  —  35  dollars,  while  B  had  ar  +  35 
dollars.     Then,  by  the  conditions, 

a:  +  35  =  3(2x-35) 
ar  +  35  =  6  a;  -  105 
-5a:  =-140. 
Whence,  x  =  28,  B's  money  at  first, 

and  2  a:  =  56,  A's  money  at  first. 

4.  What  number  is  that  whose  double  exceeds  its  half  by 
45? 

5.  Divide  34  into  two  parts  such  that  four-sevenths  of 
one  part  may  be  equal  to  two-fifths  of  the  other. 

6.  What  number  exceeds  the  sum  of  its  third,  tenth,  and 
twelfth  parts  by  58  ? 

7.  Divide  59  into  two  parts  such  that  the  sum  of  one- 
seventh  the  greater  and  one-third  the  less  shall  be  equal  to 
18. 


PROBLEMS.  121 

8.  A  is  four  times  as  old  as  B,  and  in  30  years  he  will 
be  only  twice  as  old  as  B.     What  are  their  ages? 

9.  A  is  62  years  of  age,  and  B  is  36.  How  many  years 
is  it  since  A  was  three  times  as  old  as  B? 

10.  A  had  one-half  as  much  monej'  as  B  ;  but  after  B  had 
given  him  $42,  he  had  four  times  as  much  as  B.  How  much 
had  each  at  first? 

11.  Divide  207  into  two  parts  such  that  one-fourth  the 
greater  shall  exceed  two-sevenths  the  less  by  3. 

12.  What  two  numbers  are  those  whose  difference  is  3, 
and  the  difference  of  whose  squares  is  51  ? 

13.  A  drover  paid  $1428  for  a  lot  of  oxen  and  cows. 
For  the  oxen  he  paid  $55  each,  and  for  the  cows  $32  each ; 
and  he  has  twice  as  many  cows  as  oxen.  How  many  has 
he  of  each? 

14.  Divide  80  into  two  parts  such  that  if  the  greater  is 
taken  from  62,  and  the  less  from  48,  the  remainders  are 
equal. 

15.  A  gentleman  left  an  estate  of  $1872  to  be  divided 
betweeii  his  wife,  three  sons,  and  two  daughters.  The  wife 
was  to  receive  three  times  as  much  as  either  of  the  daugh- 
ters, and  each  son  one-half  as  much  as  each  of  the  daugh- 
ters.    How  much  did  each  receive? 

16.  Divide  $70  between  A,  B,  and  C,  so  that  A*s  share 
may  be  three-eighths  of  B's,  and  C's  share  two-ninths  of 
A's. 

17.  In  a  garrison  of  2744  men,  there  are  12|-  times  as 
many  infantry  as  cavalry,  and  twice  as  many  cavalry  as 
artillery.     How  many  are  there  of  each  kind  ? 

18.  A  is  34  years  older  than  B  ;  and  he  is  as  much  above 
50  as  B  is  below  40.     Required  their  ages. 


122  ALGEBRA. 

19.  A  man  travelled  3036  miles.  He  went  four-sevenths 
as  many  miles  on  foot  as  by  water,  and  two-fifths  as  many 
miles  on  horseback  as  by  water.  How  many  miles  did  he 
travel  in  each  manner? 

20.  Divide  a  into  two  parts  such  that  m  times  the  first 
part  shall  be  equal  to  n  times  the  second. 

Let  X  =  the  first  part. 

Then,  a  —  x  =  the  second  part. 

By  the  conditions,         mx  =  n{a  —  x). 
Or,  mx  +  nx  =  an. 

"Whence,  x  =  — ^^,  the  first  part, 

w  +  n 

Therefore,  a-x  =  a ^^  =  -^HL.^  the  second  part. 

m  +  w      m  -{■  n 

21.  Divide  a  into  two  parts  such  that  m  times  the  first 
shall  be  equal  to  the  second  divided  by  n. 

22.  Find  four  consecutive  numbers  whose  sum  is  94. 

23.  Divide  43  into  two  parts  such  that  one  of  them  shall 
be  three  times  as  much  above  20  as  the  other  lacks  of  17. 

24.  Divide  $47  between  A,  B,  C,  and  D,  so  that  A  and  B 
together  may  have  S27,  A  and  C  $25,  and  A  and  D  $23. 

25.  If  a  certain  number  is  increased  by  15,  one-half  the 
result  is  as  much  below  80  as  the  number  itself  is  above 
100.     Required  the  number. 

26.  Divide  205  into  four  parts  such  that  the  second  is 
one-half  of  the  first,  the  third  one-third  of  the  second,  and 
the  fourth  one-fourth  of  the  third. 

27.  Eleven  years  ao^o,  A  was  4  times  as  old  as  B,  and  in 
13  years  he  will  be  only  twice  as  old.  Required  their  ages 
at  present. 

28.  Find  two  consecutive  numbers  such  that  the  differ- 
ence of  their  squares  added  to  three  times  the  greater  num- 
ber exceeds  the  less  number  bv  92. 


PROBLEMS.  123 

29.  What  number  is  that,  five-sixths  of  which  as  much 
exceeds  25  as  one-ninth  of  it  is  below  9  ? 

30.  A  is  m  times  as  old  as  B,  and  in  a  years  he  will  be  n 
times  as  old.     Required  their  ages  at  present. 

31.  Divide  a  into  three  parts  such  that  the  first  may  be  n 
times  the  second,  and  the  second  n  times  the  third. 

32.  A  can  do  a  piece  of  work  in  8  days  which  B  can 
perform  in  10  days.  In  how  many  days  can  it  be  done  by 
both  working  together  ? 

Let  X  =  the  number  of  days  required. 

Then,  -  =  what  both  can  do  in  one  day. 

X 

Also,  -  =  what  A  can  do  in  one  day, 

8 

and  —  =  what  B  can  do  in  one  day. 

By  the  conditions,   -  H —  =  — 
8     10     ar 

6x-H4a-  =  40  ' 

9x  =  40. 

Whence,  ar  =  4f,  the  number  of  days  required. 

33.  A  can  do  a  piece  of  work  in  15  days,  and  B  can  do 
the  same  in  18  days.  In  how  many  days  can  it  be  done  by 
both  working  together  ? 

34.  A  can  do  a  piece  of  work  in  3J  hours  which  B  can 
do  in  2 J  hours,  and  Q  \\i2\  hours.  In  how  many  hours  can 
it  be  done  by  all  working  together? 

35.  The  stones  which  pave  a  square  court  would  just 
cover  a  rectangular  area  whose  length  is  6  yards  longer, 
and  breadth  4  yards  shorter,  than  the  side  of  the  square. 
Required  the  area  of  the  court. 

36.  A,  B,  and  C  found  a  sum  of  money.  It  was  agreed 
that  A  should  receive  $15  less  than  one-half,  B  $13  more 
than  one-fourth,  and  C  the  remainder,  which  was  $27. 
How  much  did  A  and  B  receive  ? 


124  ALGEBRA. 

37.  A  can  do  a  piece  of  work  in  a  hours  which  B  can  do 
in  b  hours.  In  how  many  hours  can  it  be  don©  by  both 
working  together? 

38.  A  vessel  can  be  filled  by  three  taps ;  by  the  first 
alone  it  can  be  filled  in  a  minutes,  by  the  second  in  b 
minutes,  and  by  the  third  in  c  minutes.     In  what  time  will  it 

/  be  filled  if  all  the  taps  are  opened  ? 

39.  A  sum  of  money,  amounting  to  $4.32,  consists 
entirely  of  dimes  and  cents,  there  being  in  all  108  coins. 
How  many  are  there  of  each  kind  ? 

Let  X  =  the  number  of  dimes. 

Then,  108  —x  =  the  number  of  cents. 

Also,  10  a:  =  the  value  of  the  dimes  in  cents. 

By  the  conditions, 

10  a:  +  108 -x  =  432 
9  a:  =  324. 
Whence,  x  =  36,  the  number  of  dimes, 

and  108  —  a:  =  72,  the  number  of  cents. 

40.  A  man  has  $4.04  in  dollars,  dimes,  and  cents.  He 
has  one-fifth  as  many  cents  as  dimes,  and  twice  as  many 
cents  as  dollars.     How  many  has  he  of  each  kind? 

41.  A  man  has  3  shillings  7  pence  in  two-penny  pieces 
and  farthings ;  and  he  has  19  more  farthings  than  two- 
penny pieces.     How  many  has  he  of  each  kind? 

42.  I  bought  a  picture  for  a  certain  sum,  and  paid  the 
same  price  for  a  frame.  If  the  frame  had  cost  $  1  less,  and 
the  picture  75  cents  more,  the  price  of  the  frame  would 
have  been  only  half  that  of  the  picture.  Required  the  cost 
of  the  picture. 

43.  A  laborer  agreed  to  serve  for  36  days  on  condition 
that  for  every  day  he  worked  he  should  receive  $1.25,  and 

♦  for  every  day  he  was  absent  he  should  forfeit  50  cents.  At 
the  end  of  the  time  he  received  $17.  How  many  days  did 
he  work,  and  how  many  was  he  absent? 

Ca 


PROBLEMS.  125 

44.  A  has  $105,  and  B  $83.  After  giving  B  a  certain 
sum,  A  has  only  one-third  as  much  money  as  B.  How 
much  was  given  to  B  ? 

45.  A  has  a  dollars,  and  B  h  dollars.  After  giving  B  a 
certain  sum,  A  has  c  times  as  much  money  as  B.  How 
much  was  given  to  B  ? 

46.  A  vessel  can  be  emptied  by  three  taps ;  by  the  first 
alone  it  can  be  emptied  in  80  minutes,  by  the  second  in  200 
minutes,  and  by  the  third  in  5  hours.  In  what  time  will  it 
be  emptied  if  all  the  taps  are  opened  ? 

47.  The  second  digit  of  a  number  exceeds  the  first  by  2  ; 
and  if  the  number,  increased  by  6,  be  divided  by  the  sum 
of  its  digits,  the  quotient  is  5.     Required  the  number. 

Let  X  =  the  first  digit. 

Then,  x  +  2  =  the  second, 

and  2  X  +  2  =  the  sum  of  the  digits. 

The  number  itself  is  equal  to  10  times  the  first  digit,  plus  the  second, 
which  is  lOx  +  X  +  2,  or  11 X  +  2.     Hence,  by  the  conditions, 


llx  +  2  +  6_5 

2x+2 

llx  +  8  =  10x  +  10. 

Whence, 

x  =  2. 

Therefore, 

11  a:  +  2  =  24,  the  number  required. 

48.  The  first  digit  of  a  number  exceeds  the  second  by  4  ; 
and  if  the  number  be  divided  by  the  sum  of  its  digits,  the 
quotient  is  7.     Required  the  number. 

49.  The  first  digit  of  a  number  is  three  times  the  second  ; 
and  if  the  number,  increased  by  3,  be  divided  by  the  differ- 
ence of  its  digits,  the  quotient  is  16.     Required  the  number. 

50.  A  merchant  has  grain  worth  9  shillings  per  bushel, 
and  other  grain  worth  13  shillings  per  bushel.  In  what  pro- 
portion must  he  mix  40  bushels,  so  that  the  mixture  may  be 
worth  10  shillings  per  bushel? 


126  -  ALGEBRA. 

61.  Gold  is  19J  times  as  heav}'  as  water,  and  silver  lOl. 
times.  A  mixed  mass  weighs  4160  ounces,  and  displaces 
250  ounces  of  water.  How  many  ounces  of  each  metal 
does  it  contain  ? 

62.  The  second  digit  of  a  number  exceeds  the  first  by  3  ; 
and  if  the  number,  diminished  by  9,  be  divided  by  the  sum 
of  its  digits,  the  quotient  is  3.     Required  the  number. 

63.  Two  persons,  A  and  B,  63  miles  apart,  start  at  the 
same  time  and  travel  towards  each  other.  A  travels  4  miles 
an  hour,  and  B  3  miles  an  hour.  How  far  will  each  have 
travelled  when  they  meet? 

Let  X  =  the  distance  A  travels. 

Then,  63  —  a:  =  the  distance  B  travels. 

Also,  -  =  the  time  A  takes  to  travel  x  miles, 

4 

and  — ^— ^  =  the  time  B  takes  to  travel  x  miles. 


By  the  conditions. 


3 

X     63 


4  3 

3a:  =  252-4a: 

7x  =  252.  ^       - 

"Whence,  x  =  36,  the  distance  A  travels, 

and  63  —  a:  =  27,  the  distance  B  travels. 

54.  A  person  has  4^  hours  at  his  disposal.  How  far  can 
he  ride  in  a  coach  which  travels  5  miles  an  hour,  so  as  to 
return  home  in  time,  walking  back  at  the  rate  of  3 J  miles  an 
hour? 

55.  A  courier  who  travels  a  miles  daily  is  followed  after 
n  days  by  another,  who  travels  6  miles  daily.  In  how  many 
days  will  the  second  overtake  the  first  ? 

56.  Two  men,  A  and  B,  26  miles  apart,  set  out,  B  30 
minutes  after  A,  and  travel  towards  each  other.  A  travels 
3  miles  an  hour,  and  B  4  miles  an  hour.  How  far  will  each 
have  travelled  when  they  meet  ? 


^, 


'Ai 


probli:ms.  1^7 

57.  A  capitalist  invests  |  of  a  certain  sum  in  5  per  cent 
bonds,  and  the  remainder  in  6  per  cent  bonds  ;  and  finds 
that  his  annual  income  is  $180.  Required  the  amount  in 
each  kind  of  bond. 

58.  What  principal  at  r  per  cent  interest  will  amount  to 
a  dollars  in  t  years  ? 

59.  In  how  many  years  will  p  dollars  amount  to  a  dollars, 
at  r  per  cent  interest  ? 

60.  Separate  41  into  two  parts  such  that  one  divided  by 
the  other  may  give  1  as  a  quotient  and  5  as  a  remainder. 

Let  X  —  the  divisor. 

Then,  41  —  x  =  the  dividend. 

By  the  conditions,  ^^~^  =  1  +  - 

X  X 

41— x  =  T+6 
-  2  X  =  -  36. 
Whence,  x  =  18,  the  divisor,  /C/ 

and  41  -  X  =  23,  the  dividend.  — 

61-  Separate  37  into  two  parts  such  that  one  divided  by 
the  other  may  give  3  as  a  quotient  and  1  as  a  remainder. 

62.  Separate  113  into  two  parts  such  that  one  divided  by 
the  other  may  give  2  as  a  quotient  and  20  as  a  remainder. 

63.  A  general,  arranging  his  men  in  a  solid  square,  finds 
he  has  21  men  over.  But  attempting  to  add  1  man  to  each 
side  of  the  square,  he  finds  he  wants  200  men  to  fill  up  the 
square.  Required  the  number  of  men  on  a  side  at  first, 
and  the  whole  number  of  troops. 

64.  Separate  a  into  two  parts  such  that  one  divided  by 
the  other  ma}^  give  6  as  a  quotient  and  c  as  a  remainder. 

65.  The  denominator  of  a  fraction  exceeds  the  numerator 
by  6  ;  and  if  8  is  added  to  the  denominator,  the  value  of 
the  fraction  is  \.     Required  the  frnction. 


128 


ALGEBRA. 


66.  The  sum  of  the  digits  of  a  number  is  6,  and  the  num- 
ber exceeds  its  first  digit  bj'  46.     What  is  the  number? 

67.  At  what  rate  of  interest  will  p  dollars  amount  to  a 
dollars  in  t  years  ? 

68.  A  man  bought  a  picture  for  a  certain  price,  and  paid 
three-fourths  the  same  amount  for  a  frame.  If  the  frame 
had  cost  $2  less,  and  the  picture  60  cents  more,  the  price  of 
the  frame  would  have  been  one-third  that  of  the  picture. 
How  much  did  each  cost? 

69.  The  denominator  of  a  fraction  exceeds  the  numerator 
by  1.  If  the  denominator  be  increased  by  2,  the  resulting 
fraction  is  less  by  unity  than  twice  the  original  fraction. 
Required  the  fraction. 

70.  At  what  time  between  3  and  4  o'clock  are  the  hands 
of  a  watch  opposite  to  each  other? 

Let  OM  and  OH  represent  the  positions  of  the  minute  and  hour- 
M  hands  at  3  o'clock,  and  OM'  and  OH'  their 

positions  when  opposite  to  each  other. 

Let  X  =  the  arc  MHH'M'  over  which  the 
minute-hand  has  passed  since  3  o'clock. 

Then,  —  =  the  arc  HW  over  which  the  houB 
'  12 
hand  has  passed  since  -3  o'clock. 

Also,  the  arc  MH  —  15  minute-spaces, 
and  the  arc       WM'  =  30  minute-spaces. 

arc  MHH'M'  =  arc  MH+  arc  H'M'  +  arc  HH'. 

^=15+30 -f:^ 
12  ar  =  540  -f  ar 
llx  =  540. 

a:  =  49^. 

Hence  the  required  time  is  49^  minutes  after  3  o'clock. 

71.  At  what  time  between  7  and  8. are  the  hands  of  a 
watch  opposite  to  each  other? 


Now, 
That  is, 

"Whence, 


PROBLEMS.  129 

72.  At  what  time  between  2  and  3  are  the  hands  of  a 
watch  opposite  to  each  other  ? 

73.  At  what  time  between  5  and  6  are  the  hands  of  a 
watch  together? 

74.  At  what  time  between  1  and  2  are  the  hands  of  a 
watch  together  ? 

76.  A  woman  sells  half  an  egg  more  than  half  her  eggs. 
Again  she  sells  half  an  egg  more  than  half  her  remaining 
eggs.  A  third  time  she  does  the  same ;  and  now  she  has 
sold  all  her  eggs.     How  many  had  she  at  first? 

76.  A  man  has  two  kinds  of  money,  dimes  and  half- 
dimes.  If  he  is  offered  $1.35  for  20  coins,  how  many  of 
each  kind  must  he  give  ? 

77.  A  man  has  a  hours  at  his  disposal.  How  far  can  he 
ride  in  a  coach  which  travels  b  miles  an  hour,  so  as  to  return 
home  in  time,  walking  back  at  the  rate  of  c  miles  an  hour? 

78.  At  what  time  between  6  and  6.30  o'clock  are  the 
hands  of  a  watch  at  right  angles  to  each  other  ? 

79.  At  what  times  between  10  and  11  o'clock  are  the 
hands  of  a  watch  at  right  angles  ? 

80.  A  banker  has  two  kinds  of  money.  It  takes  a  pieces 
of  the  first  kind  to  make  a  dollar,  and  b  pieces  of  the  sec- 
ond kind.  If  he  is  ofiered  a  dollar  for  c  pieces,  how  many 
of  each  kind  must  he  give  ? 

81.  A  alone  can  perform  a  piece  of  work  in  12  hours ;  A 
and  C  together  can  do  it  in  5  hours ;  and  C's  work  is  two- 
thirds  of  B's.  The  work  must  be  completed  at  noon.  A 
commences  work  at  5  a.m.  ;  at  what  hour  can  he  be  relieved 
by  B  and  C,  and  the  work  be  just  finished  in  time? 

82.  At  what  time  between  4  and  5  is  the  minute-hand  of 
a  watch  exactly  5  minutes  in  advance  of  the  hour-hand  ? 


130  ALGEBRA. 

83.  A  man  buys  a  certain  number  of  eggs  at  the  rate  of 

3  for  10  cents.  He  sells  one-third  of  them  at  the  rate  of  2 
for  7  cents,  and  the  remainder  at  the  rate  of  4  for  15  cents  ; 
and  makes  16  cents  by  the  transaction.  How  many  eggs 
did  he  buy  ? 

84.  A  merchant  increases  his  capital  annually  by  one- 
thu'd  of  it,  and  at  the  end  of  each  year  sets  aside  $2700  for 
expenses.  At  the  end  of  four  years,  after  deducting  the 
amount  for  expenses,  he  finds  that  his  capital  is  reduced  to 
$2980.     What  was  his  capital  at  first? 

86.  A  man  owns  a  harness  valued  at  $25,  a  horse,  and  a 
carriage.  The  harness  and  carriage  are  together  worth  two- 
thirds  the  value  of  the  horse,  and  the  horse  and  harness  are 
together  worth  $15  more  than  twice  the  value  of  the  carriage. 
Required  the  value  of  the  horse,  and  of  the  carriage. 

86.  Two  men,  A  and  B,  107  miles  apart,  set  out  at  the 
same  time  and  travel  towards  each  other.  A  travels  at  the 
rate  of  13  miles  in  5  hours,  and  B  at  the  rate  of  11  miles  in 

4  hours.     How  far  will  each  have  travelled  when  they  meet? 

87.  A  mixture  is  made  of  a  pounds  of  coffee  at  m  cents 
a  pound,  h  pounds  at  n  cents,  and  c  pounds  at  p  cents. 
Required  the  cost  per  pound  of  the  mixture. 

88.  A,  B,  and  C  together  can  do  a  piece  of  work  in  6 
days ;  B's  work  is  one-half  of  A's,  and  C's  work  is  two- 
thirds  of  B's.  How  many  days  will  it  take  each  working 
alone  ? 

89.  A  and  B  start  in  business,  A  putting  in  f  as  much 
capital  as  B.  The  first  year,  A  gains  $150,  and  B  loses  \ 
of  his  money.  The  next  j^ear,  A  loses  \  of  his  money,  and 
B  gains  $300 ;  and  they  have  now  equal  amounts.  How 
much  had  each  at  first? 

90.  At  what  time  between  9  and  10  is  the  hour-hand  of 
a  watch  exactly  one  minute  in  advance  of  the  minute-hand  ? 


PROBLEMS.  131 

91.  A  and  B  together  can  do  a  piece  of  work  in  1^  days, 
A  and  C  in  IJ  days,  and  B  and  C  in  2f  days.  How  many 
days  will  it  take  each  working  alone? 

92.  A  man  buys  two  pieces  of  cloth,  one  of  which  con- 
tains 3  yards  more  than  the  other.  For  the  larger  piece  he 
pays  at  the  rate  of  $5  for  6  yards,  and  for  the  other  at  the 
rate  of  $  7  for  5  yards.  He  sells  the  whole  at  the  rate  of  3 
yards  for  $4,  and  makes  $8  by  the  transaction.  How  many 
yards  were  there  in  each  piece  ? 

93.  A  gentleman  distributing  some  money  among  beggars, 
found  that  in  order  to  give  them  a  cents  each,  he  should 
need  b  cents  more.  He  therefore  gave  them  c  cents  each, 
and  had  d  cents  left.     Required  the  number  of  beggars. 

94.  A  man  let  a  certain  sum  for  3  years  at  5  per  cent 
compound  interest ;  that  is,  at  the  end  of  each  year  there 
was  added  -^  to  the  sum  due.  At  the  end  of  the  third 
year  there  was  due  him  $2315.25.     Required  the  sum  let. 

95.  A  man  starts  in  business  with  $4000,  and  adds  to  his 
capital  annually  one-fourth  of  it.  At  the  end  of  each  year 
he  sets  aside  a  fixed  sum  for  expenses.  At  the  end  of 
three  years,  after  deducting  the  fixed  sum  for  expenses,  his 
capital  is  reduced  to  $2475.    What  are  his  annual  expenses  r 

96.  A  man  invests  one-third  of  his  money  in  3^  per  cent 
bonds,  two-fifths  in  4  per  cent  bonds,  and  the  balance  in  4J 
per  cent  bonds.  His  income  from  the  investments  is  $595. 
What  is  the  amount  of  his  property  ? 

97.  At  what  time  between  8  and  9  o'clock  is  the  minute- 
hand  of  a  watch  exactly  35  minutes  in  advance  of  the  hour- 
hand? 

98.  A  fox  is  pursued  by  a  greyhound,  and  has  a  start  of 
60  of  her  own  leaps.  The  fox  makes  3  leaps  while  the 
greyhound  makes  but  2  ;  but  the  latter  in  3  leaps  goes  as  far 
as  the  former  hi  7.  How  many  leaps  does  each  make  before 
the  greyhound  catches  the  fox  ? 


132  ALGEBRA. 

XIV.   SIMPLE  EQUATIONS. 

CONTAINING  TWO   UNKNOWN  QUANTITIES. 

180.  If  we  have  a  simple  equation  containing  two  un- 
known quantities,  as  x-\-y  —  12^  it  is  impossible  to  deter- 
mine the  values  of  x  and  y  definitely ;  because,  if  any  value 
be  assumed  for  one  of  the  quantities,  we  can  find  a  corre- 
sponding value  for  the  other. 

Thus,  if  a;  =9,  then  9 +  2/ =12,  or  2/=  3; 

if  flj  =  8,  then  8  -f  2/  =  12,  or  2/  =  4  ;  etc. 

Hence,  any  of  the  pau's  of  values, 

a;  =  9,2/  =  3;  a;  =  8,2/  =  4;  etc., 

will  satisfy  the  given  equation. 

Similarly,  the  equation  cc  —  2/  =  4  is  satisfied  by  any  of 
the  following  pairs  of  values  : 

a;  =  9,2/  =  5;  a;  =  8,  2/=4;  etc. 

Equations  of  this  kind  are  called  indeterminate. 

But  suppose  we  are  required  to  find  a  pair  of  values 
which  will  satisfy  both  x+y=^12  and  a;  —  2/  =  4  at  the  same 
time.     It  is  evident  by  inspection  that  the  values 

a;  =  8,  2/  =  4 

satisfy  both  equations  ;  and  no  other  pair  of  values  can  be 
found  which  will  satisfy  both  simultaneously. 

181.  Simultaneous  Equations  are  such  as  are  satisfied  by 
the  same  values  of  their  unknown  quantities. 

Independent  Equations  are  such  as  cannot  be  made  to 
assume  the  same  form. 


SIMPLE  EQUATIONS.  133 

Thus,  x  +  y  =  9  and  x  —  y=l  are  independent  equations. 

But  x-\-y=9  and  2a; -f  22/ =  18  are  not  independent, 
since  the  first  equation  may  be  obtained  from  the  second  by 
dividing  each  term  by  2. 

182.  It  is  evident  from  Art.  180  that  two  unknown  quan- 
tities require  for  their  determination  two  independent,  simul- 
taneous equations. 

Two  such  equations  may  be  solved  by  combining  them  so 
as  to  form  a  single  equation  containing  but  one  unknown 
quantity.     This  operation  is  called  Elimination. 

183.  There  are  three  principal  methods  of  elimination : 

1.  By  Addition  or  Subtraction. 

2.  By  Substitution. 

3.  By  Comparison. 

ELIMINATION  BY  ADDITION  OR  SUBTRACTION. 

184.  1.  Solve  the  equations  |  5^'  -  ^2/  =  19  (1) 

I  7a; +  42/=    2  (3) 

Multiplying  (1)  by  4,  20  a;-  122/=  76 

Multiplying  (2)  by  3,  21x  +  l2y=:    6 

Adding  these  equations,  41  a;  =  82 

Whence,  a;  =    2 

Substituting  the  value  of  a;  in  (1),  10— 32/  =  19 

-32/=    9 
Whence,  y  =  —  3 

Ans.  a;=  2,  y  =  —  3. 

This  solution  is  an  example  of  elimination  by  addition. 


i4                                     ALGEBRA. 

2.  Solve  the  equations 

1           (1) 

.24           (2) 

Multiplying  (1)  by  2, 

30a;  +  16?/  = 

2           (3) 

Multiplying  (2)  by  3, 

30a;-21^  =  - 

-72           (4) 

Subtracting  (4)  from  (3), 

37y  = 

74 

2/  = 

2 

Substituting  this  value  in  (2) 

,     10a;-14  =  - 

•24 

10fl;  =  - 

•10 

X  —  — 

•    1 

Ans.  £c  =  — 

■    l,2/  =  2. 

This  solution  is  an  example  of  elimination  by 

subtraction. 

RULE. 

Multiply  tJie  given  equations  by  such  numbers  as  will  make 
the  coefficients  of  one  of  the  unknown  quantities  equal.  Add 
or  subtract  the  resulting  equations  according  as  the  equal 
coefficients  have  unlike  or  like  signs. 

Note.  If  the  coefficients  which  are  to  be  made  equal  are  prime  to 
each  other,  each  may  be  used  as  the  multiplier  for  the  other  equation. 
If  they  are  not  prime,  such  multipliers  should  be  used  as  will  produce 
their  lowest  common  multiple. 

Thus,  in  Ex.  1,  to  make  the  coefficients  of  y  equal,  we  multiply  (1) 
by  4,  and  (2)  by  3.  But  in  Ex.  2,  to  make  the  coefficients  of  x  equal, 
since  the  L.C.M.  of  15  and  10  is  30,  we  multiply  (1)  by  2,  and  (2)  by  8. 

EXAMPLES. 

Solve  the  following  by  the  method  of  addition  or  subtrac- 
tion: 

3     <7x  +  2y==31.  ^     (2x-3y=   4. 

\3x-4:y  =  2S.  '   {ex-    3/=28. 

^     (3x-i-7y  =  33.  ^     (7y-5x  =  -n. 

1    x  +  2y=10.  '   \l5x-Uy  =  82. 


SIMPLE  EQUATI0:N^S.  135 


-ll?/  =  -58. 
2. 


8 


\Sx-{-2y=       3.  *   1  15a;-f   8y== 

I    907-13?/=    76.  jg     (ll2/-18a;=        2. 

U5a;+    42/  =  101.  *    l24a;-    52/  =  -22. 

g^    (24a;-M3y  =  -27.  ^^    |  24a;- 18?/ =  -43. 

l36a;+ll2/  =  -15.  *    U2ic  +  30^=      17. 

^Q     (152/-    8a;=12.  ^g    |lla;-122/  =  -32. 

l25t/  +  12a;=    1.  '   lll2/-12ic=      14. 

11.    f5^-72/=15.  ^Q     r    dx-ny=      24. 

l3a;-52/=13.  '   llOa;+   9y  =  -37. 

^^    I  12a; +  21 2/ =  -23. 
1  15a; +  282/ =  —  30. 

ELIMINATION   BY   SUBSTITUTION. 

185.   1.  Solve  the  equations     j  7^  -  ^2/  =  -  62  (1) 

l2y-ox=     44  (2) 

Transposing  5a;  in  (2),  2y  =  5a;  +  44 

Or,  2/  =  ^^      (3) 

Substituting  this  in  (1) ,  7x-sf^^^±^\  =  -    62 

Or,  ,  ^^_15a;+132^_ 

2 
Clearing  of  fractions,      14a;— 15a;— 132  =  —  124 

—  x=     8 
Whence,  a;  =  —  8 

Substituting  this  value  in  (3) ,  t/  =  -^^+^^  ^  2 

Ans.  a;  =  —  8,  2/ =  2. 


136  ALGEBRA.. 

RULE. 

Find  the  value  of  one  of  the  unknown  quantities  in  terms 
of  the  other  from  one  of  the  given  equations,  and  substitute 
this  value  for  that  quantity  in  the  other  equation. 

EXAMPLES. 
Solve  the  following  by  the  method  of  substitution : 

Q    J  ojo-\-ly  =  —  19. 


-   7y=     i 

4:y  —  15x  =  —  7. 


Q    jlOa;^    7y=     9 
1    4v  — 15x  = 


10    I    ex-5y  =  ^7, 
llOa;  +  32/=    11. 


g     (7x  —  2y^     8.  jj     (9x  +  2y=:15. 

'  l82/~5i«  =  -9.  '    Uaj-fV2^=   8.  _ 

^g     r   9a;-42/  =  -4.  ^g     (  8a;+    72/  =  -23. 

\    *  ll5a;-f 82/  =  -3.  '   1  5y- 12a;  =  - 12. 

-     (2iC-72/==    8.  j  jg     (72/ --3a;  =139. 

Xiy-dx^ld.  I  '  l2a;  +  5y=   91. 


ELIMINATION  BY  COMPARISON. 

186.   1.  Solve  the  equations  j2aJ-52/  =  -16 

\Bx  +  7y=       5 

Transposing  —  5yin  (1),  2x  =  5y  —  16 

Or,  «  =  5jiZLL6        (3) 

Transposing  72/  in  (2),  3x=z6  —  7y 


0) 

(2) 


SIMPLE  EQUATIONS.  187 

^        .        ,  ,  .,  5V-16      5-7y 

Equating  these  values  of  a?,         -^— —  =  — r-^ 

Clearing  of  fractions,  ISy  -  48  =  10  -  14y 

29  2/ =  58 

Substituting  this  value  In  (3) ,  x— r —  =  —  8 

^715.  a;  =  — 3,  y  =  2. 

RULE. 

i^nc?  ^fee  va?ue  o/  ^^e  same  unknown  quantity  in  terms  of 
the  other  from  each  of  the  given  equations^  and  place  these 
values  equal  to  each  other. 

EXAMPLES. 

Solve  the  following  by  the  method  of  comparison : 

g     (5x-\-Gy=      24. 
(92/-8a;  =  -26. 

g     (7x-   8y  =  -n. 
1    a;-122/=      12. 

jQ     C    5x-12y=7. 
(lOo;-    92/  =  4. 

^j     (72/-12a;=17. 
1  8a;  4- 11?/  =  20. 

j2    I    7x  +  Sy  =  e. 

lnx  + 


9y  =  8. 


jg    J  ii>x-\-6y  =  —  7. 
^  ~        21a;=«    18. 


ri55 

(8y 


138 


ALGEBRA. 


MISCELLANEOUS  EXAMPLES. 

187.    Before  applying  either  method  of  elimination,  each  of 
the  given  equations  should  be  reduced  to  its  simplest  form. 

1.   Solve  the  equations 

C     7  3 


=  0 


3a;-9  =  0,  or  ly-Zx  = 


13 
19 


From  (1), 

72/+ 28 
From  (2), 

xy  —  ^x  —  xy  -\-  b y  =  ^  13,  or  by  —  2x  —  —  \^ 
Multiplying  (3)  by  2,  Uy-Qx  =  -  38 

Multiplying  (4)  by  3,  151/  -  6  a;  =  -  39 

Subtracting  (5)  from  (6),  2/  =  —    1 


Substituting  in  (4) , 


Solve  the  following ; 
'll?/  +  6a;=115. 

2x  _  ii_^  _  _  5. 

3         6  2* 


-5-2a;=-13 
-2a;  =  -   8 
a;  =  4. 
Ans.  a;  =  4,  2/  =  —  1. 


3. 


+  3y  =  -46. 


^+3a;=66. 

2        3 

^  +  ^  =  -4. 
L8^6 


[5      6      90* 


(1) 
(2) 

(3) 

(4) 
(5) 
(6) 


.2a; -.05  2/ =.25. 
.03  a; +  .32/ =.96. 

.5a;+  22/=     .01. 
.lla;+.32/  =  -.009. 


SIMPLE  EQUATIONS. 


189 


8. 


2  3 

3  4 


9. 


Q      2x-^Sy^y 
5  3 

4:y  —  3x__3x  .  . 
6        ""4: 


a;(22/-3)  =  22/(a;+l), 
10.    ^      3     ^_5_,^o. 


x-l      2/  +  2 


11. 


a^(2/-3)-y(«  +  4)  =  22. 

(y  +  l)(a:-2)-(y  +  3)(a;-4)  =  6. 


12. 


'  x  +  y      5 
flj-y     3 


16. 


2a;  +  3y   ^1, 
a:  +  y  +  13  2* 

5a;      7y-2_j^^ 


18. 


-12 


^  +  8. 
4 


x  +  y  _2y-x__^^ 

5  4 


16. 


—  St.  2—  ^^ 

6  10 


2y  — 3      5  — 3a?_ 


y- 


14. 


y- 


3a;  +  2^y  +  2 
5  3 

2y  +  l_a;  — 6 


17.    ^ 


a;  +  3y^     3 
2a;  — 2/         8* 


7y  —  x 
24-a;  +  2y 


=  -17. 


18. 


x  —  5      2a;  — y  — l_2y  — 2 
4  3  5      ' 

2y  +  a;  — l_a;  +  y 
9  4 


140 


ALGEBRA. 


19. 


3x  _y     X     2y 
T~3      2      7" 


13 
4 


41/  — 3a;=  11. 


'2^x-by      3^2a;  +  y 
2        "^  5      ' 

^__x-2y  ^x     y 
4  2      8 


21. 


^      2a;  +  2/^17      2y  +  a; 
3  12  4 


5      2a;  — y_         2y  — a? 
4  4      "^  3      * 


22. 


2^  __  3^  __  a;  +  2i/  _  q  __  5a?  — 6y^ 
3        5  4      ■"  4       * 

2^^  5  ^15 


a?-2y 


3aj 


2a;  — 42/-1      6a;-l 
3_5w      4a;-13 

0» il  —  . 


a;  +  2 


'4ar'4-4a;y  +  272=  (a;  +  2/)(4a5+17). 

y(a;-i/)  +  54^5y  +  27^ 
x  —  y  5 


25. 


0^  _  42,2  _17  =  (a;  4- 22^  -  2)  (a;  -  22/ +1) . 

xy  —  b  ,  1  —  2  a; 
2^-2        y-1 


SIMPLE  EQUATIONS. 


141 


Note.   In  solving  literal   simultaneous   equations,  the   method  of 
elimination  by  addition  or  subtraction  is  usually  to  be  preferred. 


26.    Solve  the  equations 

(    ax-\-by  =  c 
\a'x  +  b'y=:e' 

(1) 
(2) 

Multiplying  (1)  by  6', 

ab 

x  +  bb'y=b'c 

Multiplying  (2)  by  b, 

a'bx  +  bb'y  =  be' 

Subtracting, 

ab 

X  —  a'bx  =  b'c  —  be' 

Whence, 

b'c-bd 
^     ab'  -  a'b 

Multiplying  (1)  by  a', 

aa'x  +  a'by  =  a'c 

(3) 

Multiplying  (2)  by  a, 

aa'x  -f  ab'y  =  ac' 

(4) 

Subtracting  (3)  from  (4) , 

ab 

'y  —  a'by=  ac'—a'c 

Whence, 

ac'  —  a'c 
^     ab'-a'b 

Solve  the  following  equations : 

2^      (2x-3y  =  a. 
XSx  +  Ayz^b. 

33.    ^ 

a     b 

2g      (  ax  +  by  =  m. 

1+5  =  "- 
,c     a 

29.    1  ^^  —  ^2/  =  c. 
1    a;—   y  =  d. 

-f 

'  x-\-ay=ia{a 

+  26). 

30.  1  aa;-6y  =  0. 
1  mx-\-ny=p. 

31.  |«^  +  ^2/  =  '^- 
\  ex  —  dy  =  n. 

32.  \'^^-f^y=P' 
\  m'x  —  n'y  =: p' . 

^- 

86. 

'  ax -{-by  =  2. 
.ab{ay  —  bx): 

a     0 

+  6) 

-6^. 

142 


37. 


mx-^ny 


nx-\-my 


ALGEBRA. 

mn 


36. 


(a  +  h)x  —  (a  —  h)y  =  4a&. 
(a-6)a?-(a  +  6)2/  =  0. 


ha  ah 

x  —  h  _  y  —  a  _  (J?  —  h^ 
a  h  ah 


■\ 


X  y     _a^-{-m^—n^ 

40.    ^  ot     m-|-w~   a(mH-n) 

(m  -\-ny{m  —  n)x  =  a^y. 


41. 


-^4--l-=— i— .  f      X  y 

a^h^a-h     a'-h'  hri:7.  +  -^=^^- 

42.    J  ^  +  ^      a  —  0 


^_+_l___J_ 


^a  —  6     a  +  h     a^ — 


fl?  — y  =  4a6. 


Note.  Certain  fractional  equations,  in  which  the  unknown  quanti- 
ties occur  in  the  denominators,  are  readily  solved  without  previously 
clearing  of  fractions. 


43.    Solve  the  equations 


i2-£=  8 

X      y 
X      y 


(1) 
(2) 


Multiplying  (1)  by  5,        ^-15=   40 

Multiplying  (2)  by  3,       ^i  +  l^^-S 

X       y 


Adding, 


Z*=   37 


SIMPLE  EQUATIONS. 


148 


37  x=    74 
x=      2 


Substituting  in  (1), 


5-2  = 


y 
y 

2/  =  -3 
Ans.  a;  =  2,  y  =  — 3. 


Solve  the  following  equations  : 


44. 


X     y 

2_3 

La;     y 


48. 


\    X         V  "~ 


a;      y 


45. 


?-?  = 

X     y 

15__8 
X      y 


17 
3* 


I  ^    y 


c      d 

-  -f  -  =3  n. 

X     y 


46. 


11 

X 


2/      2' 


X     y 


50. 


r_2__ 

9a;'   2y 


=  -3. 


5-  +  i.  =  II. 
L3a;     42/       6* 


47. 


2-^  =  16. 

a;     22/ 

2a;     y 


61. 


-^  +  -  =  ^^(^+^)' 

-  +  —  =  wi*  -H  n*. 
X     y 


144  ALGEBRA. 


XV.    SIMPLE    EQUATIONS. 

CONTAINING  MORE  THAN  TWO  UNKNOWN  QUANTITIES. 

188.  If  there  are  thi-ee  simple  equations  containiDg  three 
unknown  quantities,  we  may  combine  two  of  them  by  the 
methods  of  elimination  explained  in  the  last  chapter,  so  as 
to  obtain  an  equation  containing  onl}"  two  unknown  quanti- 
ties. We  may  then  combine  the  third  equation  with  either 
of  the  others,  and  obtain  another  equation  containing  the 
same  two  unknown  quantities.  By  solving  the  equations 
thus  obtained,  we  derive  the  values  of  two  of  the  unknown 
quantities.  These  values  being  substituted  in  either  of  the 
given  equations,  the  value  of  the  third  unknown  quantity 
may  be  determined. 

A  similar  method  may  be  used  when  the  number  of  equa- 
tions and  of  unknown  quantities  is  greater  than  three. 

The  method  of  elimination  by  addition  or  subtraction  is 
usually  the  most  convenient. 

189.  1.  Solve  the  equations 

Qx-    4?/-    7z=      17  (1) 

2x-    ly-Uz^     29  (2) 

10a;-   5y-   3z==     23  (8) 

Multiplying  (1)  by  3,    18a;- 122/- 21 2=     51 
Multiplying  (2)  by  2,    18a;  -  Uy  -  322;  =     58 

Subtracting,  2y  +  nz  =  -    7  (4) 

Multiplying  (1)  by  5,    30a;  -  20y  -3oz=     85  (5) 

Multiplying  (3)  by  3,    S0x-16y-    9z=     69  (6) 

Subtracting  (5)  from  (6),         5y-\-26z  =  -U  (7) 


SIMPLE  EQUATIONS.  145 


Multiplying  (4)  by  5, 

10?/ +  552!  =  -35 

Multiplying  (7)  by  2, 

10?/ 4- 522;  =  -32 

Subtracting, 

2>z  =  -   3 
z  =  —    1 

Substituting  in  (4) , 

22/-ll  =  -    7 

.-.2/=        2 

Substituting  the  values  of  y 

and  2;  in  (1), 

6a;-8  +  7=      17 

.•.a;=        3 

Ans.  x  =  3,y  =  2,  z 

'u+x+y=   6 

(1) 

x  +  y+z^'   7 

(2) 

y+2+«=   8 

(3) 

L  z  +  u  +  x=    9 

(4) 

-1. 

In   certain  cases  the  solution  may  be  abridged  by  aid  of 
the  artifice  which  is  employed  in  the  following  example. 


2.  Solve  the  equations 


Adding  the  given  equations, 

3w  +  3a;  +  32/  +  32;  =  30 
Whence,  u  +  x-\-y-\-z  =  10  (5) 

Subtracting  (2)  from  (5) ,  u=    S 

Subtracting  (3)  from  (5),  x=    2 

Subtracting  (4)  from  (5),  2/=    1 

Subtracting  (1)  from  (5),  2=4 

EXAMPLES. 
.  Solve  the  following  equations  : 

x-^y=      2.  (2x  —  5y=  —  19. 


3.    ^2/  +  2  =  -l-  4.    ■l3y-\-4:Z=      13. 

z4-x=     3.  (22:-5aJ=      12. 


146 


10 


ALGEBRA. 

3a;-2?/  =  -l. 

r7ir  +  4?/—    2;  =  — 50 

5^  +  42  =  — 6. 

13. 

)^x  —  by  —  ^z=      20 

x  —  y  —  3z  =  ll. 

(    a;  — 3?/  — 42!=      30 

2x-    y  =  5. 
ZX'\-2y—    2;  =  6. 
x  —  ^y-\-2z=l. 

14. 

(    a;-62/  +  42=    3. 
3  4a;  +  4y-32!=10. 
L2X+    2/  +  62!  =  46. 

■■{' 

Kx 


x-\-    y+    z—    53. 

-{-2y-\-Sz=107. 

a;  +  3y+42=137. 


Sx—    y  —  2z  =  ~23. 

8.    Hx-{-2y  +  3z=      15. 

4:X+3y—    z  =  —    6, 


aj-fy  — 2;  =  8. 
y +  2;  — a;=  1. 
2;  +ic  — 2/  =  —  11. 


ra;— 22/H-32;=        0 

.    jy  —  2z-hSx  =  —  26 

Lz—2x-\-3y=       9 


r6a;-32/+22!  =  41. 

11.     }2x+    y-    2  =  17. 

(5aj  +  42/-22  =  86. 


2a;+    2/+    2J  =  -2. 
12.     ^     a: +  2^+    z=      0. 

a;_|.      y4.22!  =  -4. 


16 


r    8x-9y-7z  = 

.     -J  12ir-   y-3z  = 

L  &x—2v—   2  = 


16. 


17. 


18. 


19. 


8x—9y—7z  =  —  S6. 

36. 

2y-   2=      10. 


4aj-32/  +  22  =  40. 
5a;4-92/-72!  =  47. 
937  +  82^-32  =  97. 


2      3      4 


3      4 


4      2      3 


2u-Sx 
Sx  —  4:y 
4:y  —  5z 
bz  -Qu 


-43. 

34. 

-50. 


f2y  +  2-f-2w  =  -28. 
2/  +  82  =-2. 
4»+z  =  13. 

|  +  3t^  =  -20. 
3 


SIMPLE  EQUATIONS. 


147 


20. 


21. 


23. 


24. 


■l*\ 

=  1. 

\*\ 

3 
2* 

hi 

=  2. 

r3__2 

X     y 

=  -13. 

y    ^ 

=  14. 

3      2 

^Z        X 

=  18. 

■ax  + 

DtV  =  2. 

a^2/  + 

a«2;=2. 

.a^z  +  i 

x^a;  =  a«+l 

3w  — 2;=22  — »  — 2t/. 
4ic-2/  =  35-32;. 
4^  —  2?/=  19  — 3a;. 
2  =  39-2^-42/. 


12      3 

-  +  -  +  -  =  -7. 
X     y     z 

?-?  +  f  =  9. 

X      y      z 


26. 


27. 


28. 


x  +  z 


x—y     x—z 
5  6 


=  1. 

=  0. 


L±^_^dLy  =  _4 

4  2 

ay  -f  5a;  =  c. 
ex  -\-az  =  h. 
bz  -{-cy  =  a. 


4 

3 

25-12(a;  +  z)  =  -2/. 

'2x-\-y     y-^z^i 
4  3 

a;  +  3y     a;  —  z_     p 
3  4     ~       * 


2!  +  2/     2;  +  a;_      3 

[3            4     ~      2 

»•) 

'aa;+2/— 2;  =  a2+a— 1. 
ay-{'Z—x  =  a^—a-{-l. 
^az-\-x—y=ia. 

30. 


a;  —  ay  +  a'^z  =  a*. 
x  —  hy-^-hh^  53. 
^  —  cy  +  c^z  =  c*. 


*  Add  the  equations  together. 


148  ALGEBKA. 


XVI.  PROBLEMS. 

LEADING  TO    SIMPLE    EQUATIONS    CONTAINING   MORE 
THAN  ONE  UNKNOWN  QUANTITY. 

190.  In  solving  problems  where  more  than  one  letter  is 
used  to  represent  the  unknown  quantities,  we  must  obtain 
from  the  conditions  of  the  problem  as  many  independent 
equations  as  there  are  unknown  quantities  to  be  determined. 

1.  Divide  81  into  two  parts  such  that  f  the  greater  shall 
exceed  f  the  less  by  7. 

Let  X  =  the  greater  part, 

and  y  =  the  less. 

By  the  conditions. 


/  x-t-y  =  ox 
(       5        9 


Solving  these  equations,  a;  =  45,  y  =  36. 

2.  If  3  be  added  to  both  numerator  and  denominator  of 
a  fraction,  its  value  is  -| ;  and  if  2  be  subtracted  from  both 
numerator  and  denominator,  its  value  is  ^.  Required  the 
fraction. 


Let 

X  =  the  numerator, 

id                                        y  =  the  denominator. 

'  x  +  3     2 

By  the  conditions,  - 

y  +  3     3 
x-2      1 

[y-i~2 

Solving  these  equations,  x=7,y  =  12. 

Therefore  the  f ract 

ion  is  1. 

PROBLEMS.  149 


PROBLEMS. 

3.  Divide  50  into  two  parts  such  that  three-eighths  of  the 
greater  shall  be  equal  to  two- thirds  of  the  less. 

4.  Find  two  numbers  such  that  7  times  the  greater  ex- 
ceeds 4"  the  less  by  97,  and  7  times  the  less  exceeds  -J-  the 
greater  by  47. 

5.  If  one-fifth  of  A's  age  were  added  to  two-thirds  of 
B's,  the  sum  would  be  19^  years ;  and  if  two-fifths  of  B's 
age  were  subtracted  from  seven-eighths  of  A's,  the  remain- 
der would  be  18^  years.     Required  their  ages. 

6.  If  1  be  added  to  the  numerator  of  a  certain  fraction, 
its  value  is  ^;  and  if  1  be  added  to  its  denominator,  its 
value  is  \.     Required  the  fraction. 

7.  A  gentleman  at  the  time  of  his  marriage,  found  that 
his  wife's  age  was  j  of  his  own ;  but  after  they  had  been 
married  12  years,  her  age  was  f  of  his.  Required  their 
ages  at  the  time  of  their  marriage. 

8.  A  and  B  engaged  in  trade,  A  with  $240  and  B  with 
$96.  A  lost  twice  as  much  as  B  ;  and  on  settling  their 
accounts,  it  appeared  that  A  had  three  times  as  much 
remaining  as  B.     How  much  did  each  lose? 

9.  Eight  years  ago,  A  was  4  times  as  old  as  B  ;  but  in  12 
years  he  will  be  only  twice  as  old.  Required  their  ages  at 
present. 

10.  If  5  be  added  to  both  terms  of  a  fraction,  its  value 
is  ^ ;  and  if  3  be  subtracted  from  both,  its  value  is  \. 
Required  the  fraction. 

11.  A  and  B  agreed  to  dig  a  well  in  10  days  ;  but  having 
labored  together  4  days,  B  agreed  to  finish  the  job,  which 
he  did  in  16  days.  In  how  many  days  could  each  of  them 
alone  dig  the  well? 


150  ALGEBRA. 

12.  If  the  greater  of  two  numbers  be  divided  by  the  less, 
the  quotient  is  2  and  the  remainder  12  ;  but  if  4  times  the 
less  be  divided  by  the  greater,  the  quotient  is  1  and  the 
remainder  14.     Required  the  numbers. 

13.  If  the  numerator  of  a  fraction  be  doubled,  and  the 
denominator  increased  by  7,  its  value  is  |;  and  if  the 
denominator  be  doubled,  and  the  numerator  increased  by  2, 
the  value  is  f .     Required  the  fraction. 

14.  If  a  —  1  be  subtracted  from  the  numerator  of  a  cer- 
tain fraction,  its  value  is  a  + 1  ;  and  if  a  be  added  to  its 
denominator,  its  value  is  a.     Required  the  fraction. 

15.  A  gentleman's  two  horses,  with  their  harness,  cost 
$300.  The  value  of  the  poorer  horse,  with  the  harness, 
was  $  20  less  than  the  value  of  the  better  horse  ;  and  the 
value  of  the  better  horse,  with  the  harness,  was  twice  that 
of  the  poorer  horse.     What  was  the  value  of  each? 

16.  A  merchant  has  three  kinds  of  sugar.  He  sells  3 
lbs.  of  the  first  quality,  4  lbs.  of  the  second,  and  2  lbs.  of 
the  third,  for  60  cents ;  or,  4  lbs.  of  the  first  quality,  1  lb. 
of  the  second,  and  5  lbs.  of  the  third,  for  59  cents ;  or,  1 
lb.  of  the  first  quality,  10  lbs.  of  the  second,  and  3  lbs.  of 
the  third,  for  90  cents.  Required  the  price  per  pound  of 
each  quality. 

17.  A  sum  of  money  was  divided  equally  between  a  cer- 
tain number  of  persons.  Had  there  been  3  more,  each 
would  have  received  $1  less;  had  there  been  6  less,  each 
would  have  received  $5  more.  How  many  persons  were 
there,  and  how  much  did  each  receive  ? 

Let  X  =  the  number  of  persons, 

and  y  =  what  each  received. 

Then,  xy  =  the  sum  dirided. 

By  the  conditions, 

Ux^S){y-l)  =  xy 

I  {x~6)iy-^6)  =  xy. 
Solving  these  equations,  a:  =  12,  y  =  5. 


PROBLEMS.  151 

18.  A  boy  spent  his  money  for  oranges.  If  he  had  got 
five  more  for  his  money,  they  would  have  cost  a  half -cent 
each  less ;  if  three  less,  they  would  have  cost  a  half -cent 
each  more.  How  much  money  did  he  spend,  and  how  many 
oranges  did  he  get? 

19.  A  merchant  has  two  kinds  of  grain,  worth  60  and  90 
cents  per  bushel  respectively.  How  many  bushels  of  each 
kind  must  he  take  to  make  a  mixture  of  40  bushels,  worth 
80  cents  per  bushel  ?  

20.  My  income  and  assessed  taxes  together  amount  to 
$50.  If  the  income  tax  were  increased  50  per  cent,  and 
the  assessed  tax  diminished  25  per  cent,  they  would 
together  amount  to  $52.50.  Required  the  amount  of  each 
tax. 

21.  A  man  purchased  a  certain  number  of  eggs.  If  he 
had  bought  20  more  for  the  same  money,  they  would  have 
cost  a  cent  apiece  less;  if  15  less,  a  cent  apiece  more. 
How  many  eggs  did  he  buy,  and  at  what  price  ? 

22.  If  a  certain  lot  of  laud  were  8  feet  longer  and  2  feet 
wider,  it  would  contain  656  square  feet  more  ;  and  if  it  were 
2  feet  longer  and  8  feet  wider,  it  would  contain  776  square 
feet  more.     Required  its  length  and  width. 

23.  If  B  gives  A  $5,  they  will  have  equal  amounts  ;  but 
if  A  gives  B  $  15,  B  will  have  ^  as  much  as  A.  How  much 
money  has  each  ? 

24.  Find  three  numbers  such  that  the  first  with  half  the 
other  two,  the  second  with  one-third  the  other  two,  and  the 
third  with  one-fourth  the  other  two,  may  each  be  equal  to 
34. 

26.  There  are  four  numbers  whose  sum  is  136.  Twice 
the  first  exceeds  the  second  by  46,  twice  the  second  ex- 
ceeds the  third  by  44,  and  twice  the  third  exceeds  the  fourth 
by  40.     Required  the  numbers. 


152  ALGEBRA. 

26.  The  sum  of  the  digits  of  a  number  of  three  figures  is 
13.  If  the  number,  decreased  by  8,  be  divided  by  the  sum 
of  its  second  and  third  digits,  the  quotient  is  25  ;  and  if  99 
be  added  to  the  number,  the  digits  will  be  inverted.  Re- 
quired the  number. 

Let  X  —  the  first  digit, 

y  =  the  second, 
and  z  =  the  third. 

Then,        100x-{-10y  +  z  =  the  number, 
and  100  z-\-  lOy  -\-  x=  the  number  with  its  digits  inverted. 

By  the  conditions, 

x  +  y-\-z=13, 
lOOx+lO.y  +  g-S^og 

and       100  a:  +  10^  +  ^+99=  100;?  + 10y  + a:. 
Solving  these  equations,  x  =  2,  y  z=S,  z  =  S. 
Therefore  the  number  is  283. 

27.  The  sum  of  the  digits  of  a  number  of  two  figures  is 
11 ;  and  if  27  be  subtracted  from  the  number,  the  digits  will 
be  inverted.     Required  the  number. 

28.  The  sum  of  the  digits  of  a  number  of  three  figures  is 
1 1 ,  and  the  units'  figure  is  twice  the  figure  in  the  hundreds' 
place.  If  297  be  added  to  the  number,  the  digits  will  be 
inverted.     Required  the  number. 

29.  A  and  B  can  perform  a  piece  of  work  in  6  days,  A 
and  C  in  8  days,  and  B  and  C  in  12  da^^s.  In  how  many 
days  can  each  of  them  alone  perform  it? 

30.  If  I  were  to  make  my  field  5  rods  longer  and  4  rods 
wider,  its  area  would  be  increased  by  240  square  rods ;  but 
if  I  were  to  make  its  length  4  rods  less,  and  its  width  5  rods 
less,  its  area  would  be  diminished  by  210  square  rods. 
Required  its  length,  width,  and  area. 

31.  Find  three  numbers  such  that  the  sum  of  the  first  and 
second  is  c,  of  the  second  and  third  is  a,  and  of  the  third 
and  first  is  b. 


PROBLEMS.  153 

32.  There  is  a  number  of  three  figures,  whose  digits  have 
equal  dififerences  in  their  order.  If  the  number  be  divided 
by  half  the  sum  of  its  digits,  the  quotient  is  41  ;  and  if  396 
be  added  to  the  number,  the  digits  will  be  inverted.  Re- 
quired the  number. 


33.  A  sum  of  money  is  divided  equally  between  a  certain 
number  of  persons.  Had  there  been  m  more,  each  would 
have  received  a  dollars  less  ;  if  n  less,  each  would  have 
received  6  dollars  more.  How  many  persons  were  there, 
and  how  much  did  each  receive  ? 

34.  A  gentleman  left  a  sum  of  money  to  be  divided 
between  his  four  sons,  so  that  the  share  of  the  eldest  should 
be  \  the  sum  of  the  shares  of  the  other  three,  of  the  second 
\  the  sum  of  the  other  three,  and  of  the  third  \  the  sum  of 
the  other  three.  It  was  found  that  the  share  of  the  eldest 
exceeded  that  of  the  youngest  by  S140.  What  was  the 
whole  sum,  and  how  much  did  each  receive? 

35.  A  grocer  bought  a  certain  number  of  eggs,  part  at  2 
for  5  cents  and  the  rest  at  3  for  8  cents,  and  paid  for  the 
whole  $1.71.  He  sold  them  at  36  cents  a  dozen,  and  made 
27  cents  by  the  transaction.  How  many  of  each  kind  did 
he  buy  ? 

36.  If  a  number  of  two  figures  be  divided  by  the  sum  of  its 

digits,  the  quotient  is  7 ;  and  if  the  digits  be  inverted,  the 
quotient  of  the  resulting  number,  increased  by  6,  divided  by 
the  sum  of  the  digits,  is  5.     Required  the  number. 

37.  If  45  be  added  to  a  certain  number  of  two  figures, 
the  digits  will  be  inverted ;  and  if  the  resulting  number  be 
divided  by  the  sum  of  its  digits,  the  quotient  is  7  and  the 
remainder  6.     Required  the  number. 

38.  A  and  B  can  do  a  piece  of  work  in  m  days,  B  and  C 
in  n  days,  and  C  and  A  in  p  days.  In  what  time  can  each 
alone  perform  the  work? 


154  ALGiyBRA. 

39.  A  crew  can  row  10  miles  in  50  minutes  down  stream, 
and  12  miles  in  an  hour  and  a  half  against  the  stream. 
Find  the  rate  in  miles  per  hour  of  the  current,  and  of  the 
crew  in  still  water. 

Let  X  —  the  rate  of  the  crew  in  still  water, 

and  y  =  the  rate  of  the  current. 

Then,  x  -\-  y  =  the  rate  rowing  clown  stream, 

and  X  —  y  =  the  rate  rowing  up  stream. 

Since  the  distance  divided  by  the  rate  gives  the  time,  we  have,  by 
the  conditions, 

^    10    ^5 

x-^y     6 

12  ^3 

x-y      2* 

Solving  these  equations,  a:  =  10, 3/  =  2. 

40.  A  crew  can  row  a  miles  in  h  hours  down  stream,  and 
c  miles  in  d  hours  against  the  stream.  Find  the  rate 
in  miles  per  hour  of  the  current,  and  of  the  crew  in  still 
water. 

41.  A  boatman  can  row  down  stream  a  distance  of  20 
miles,  and  back  again,  in  10  hours ;  and  he  finds  that  he 
can  row  2  miles  against  the  current  in  the  same  time  that 
he  rows  3  miles  with  it.  Required  his  time  in  going  and  in 
returning. 

42.  A  number  consists  of  three  digits  whose  sum  is  21. 
The  sum  of  the  first  digit  and  twice  the  second  exceeds  the 
third  by  8  ;  and  if  198  be  added  to  the  number,  the  digits 
will  be  inverted.     Required  the  number. 

43.  A  merchant  has  two  casks  of  wine.  He  pours  from 
the  first  cask  into  the  second  as  much  as  the  second  con- 
tained at  first ;  he  then  pours  from  the  second  into  the  first 
as  much  as  was  left  in  the  first ;  and  again  from  the  first 
into  the  second  as  much  as  was  left  in  the  second.  There 
are  now  16  gallons  in  each  cask.  How  many  gallons  did 
each  contain  at  first? 


PROBLEMS.  155 

44.  A  number  consists  of  two  figures.  If  the  digits  be 
inverted,  the  sum  of  the  resulting  number  and  the  original 
number  is  121  ;  and  if  the  number  be  divided  by  the  sum  of 
its  digits,  the  quotient  is  5  and  the  remainder  10.  Required 
the  number. 

45.  A  man  has  $30,000  invested  at  a  certain  rate  of 
interest,  and  owes  $20,000,  on  which  he  pays  interest  at 
another  rate ;  and  the  interest  which  he  receives  exceeds 
that  which  he  pays  by  $800.  Another  man  has  $35,000 
invested  at  the  second  rate  of  interest,  and  owes  $24,000, 
on  which  he  pays  interest  at  the  first  rate  ;  and  the  interest 
which  he  receives  exceeds  that  which  he  pays  by  $310. 
What  are  the  two  rates  of  interest  ? 

46.  A  certain  sum  of  money,  at  simple  interest,  amounted 
in  2  years  to  $132,  and  in  5  years  to  $150.  Required  the 
sum,  and  the  rate  of  interest. 

47.  A  certain  sum  of  money,  at  simple  interest,  amounted 
in  m  years  to  a  dollars,  and  in  n  years  to  h  dollars.  Re- 
quired the  sum,  and  the  rate  of  interest. 

48.  A  train  running  from  A  to  B  meets  with  an  acddent 
which  causes  its  speed  to  be  reduced  to  one-third  of  what  it 
was  before,  and  it  is  in  consequence  5  hours  late.  If  the 
accident  had  happened  60  miles  nearer  B,  the  train  would 
have  been  only  1  hour  late.  What  was  the  rate  of  the  train 
before  the  accident? 

Let  ^x  =  the  rate  of  the  train  before  the  accident. 

Then,  x  =  its  rate  after  the  accident. 

Let  y  =  the  distance  to  B  from  the  point  of  detention. 

By  the  conditions,  ^  =  JL  4-  5 

X     3x 

■V-60^.V-60  J  ^ 

X  Sx 

Solving  these  equations,  x  =  10. 

Hence  the  rate  of  the  train  before  the  accident  was  30  miles  an  hour. 


156  ALGEBRA. 

49.  A  man  rows  down  a  stream,  whose  rate  is  S^  miles 
per  hour,  for  a  certain  distance  in  1  hour  and  40  minutes. 
In  returning,  it  takes  him  6  hours  and  30  minutes  to  arrive 
at  a  point  2  miles  short  of  his  starting-place.  Find  the 
distance  which  he  rowed  down  stream,  and  his  rate  of  pulling. 

50.  If  a  certain  number  be  divided  by  the  sum  of  its  two 
digits,  the  quotient  is  6  and  the  remainder  1.  If  the  digits 
be  inverted,  the  quotient  of  the  resulting  number  increased 
by  8,  divided  by  the  sum  of  the  digits,  is  6.  Required  the 
number. 

51.  A  train  running  from  A  to  B  meets  with  an  accident 
which  delays  it  30  minutes  ;  after  which  it  proceeds  at  three- 
fifths  its  former  rate  and  arrives  at  B  2  hours  and  30  minutes 
late.  If  the  accident  had  occurred  30  miles  nearer  A,  the 
train  would  have  been  3  hours  late.  What  was  the  rate  of 
the  train  before  the  accident? 

52.  A,  B,  and  C  together  have  $24.  A  gives  to  B  and 
C  as  much  as  each  of  them  has  ;  B  gives  to  A  and  C  as 
much  as  each  of  them  then  has  ;  and  C  gives  to  A  and  B  as 
much  as  each  of  them  then  has.  The}^  have  now  equal 
amounts.     How  much  did  each  have  at  first? 

53.  A  and  B  are  building  a  fence  126  feet  long.  After 
3  hours,  A  leaves  off,  and  B  finishes  the  work  in  14  hours. 
If  7  hours  had  occurred  before  A  left  off ,  B  would  have 
finished  the  work  in  4|-  hours.  How  many  feet  does  each 
build  in  one  hour? 

54.  Divide  115  into  three  parts  such  that  the  first  part 
increased  by  30,  twice  the  second  part,  increased  by  2,  and 
6  times  the  third  part,  increased  by  4,  may  all  be  equal  to 
each  other. 

55.  Four  men.  A,  B,  C,  and  D,  play  at  cards,  B  having 
$1  more  than  C.  After  A  has  won  half  of  B's  money,  B 
one-third  of  C's,  and  C  one-fourth  of  D's,  A,  B,  and  C 
have  each  $18.     How  much  had  each  at  first? 


PROBLEMS.  157 

56.  A  gives  to  B  and  C  as  much  as  each  of  them  has  ;  B 
gives  to  A  and  C  as  much  as  each  of  them  then  has  ;  and  C 
gives  to  A  and  B  as  much  as  each  of  them  then  has.  Each 
has  now  $48.     How  much  did  each  have  at  first? 

57.  A,  B,  and  C,  were  engaged  to  mow  a  field.  The  first 
day,  A  worked  2  hours,  B  3  hours,  and  C  5  hours,  and 
together  they  mowed  1  acre ;  the  second  day,  A  worked  4 
hours,  B  9  hours,  and  C  6  hours,  and  all  together  mowed  2 
acres;  the  third  day,  A  worked  10  hours,  B  12  hours,  and 
C  5  hours,  and  all  together  mowed  3  acres.  In  what  time 
could  each  alone  mow  an  acre  ? 

58.  A  man  invests  $3600,  partly  in  3^  per  cent  bonds, 
and  partly  in  4  per  cent  bonds.  The  income  from  the  3^ 
per  cent  bonds  exceeds  the  income  from  the  4  per  cent 
bonds  by  $6.     How  much  has  he  in  each  kind  of  bond? 

59.  A  and  B  run  a  race  of  480  feet.  The  first  heat,  A 
gives  B  a  start  of  48  feet,  and  beats  him  by  6  seconds  ;  the 
second  heat,  A  gives  B  a  start  of  144  feet,  and  is  beaten  by 
2  seconds.     How  many  feet  can  each  run  in  a  second  ? 

60.  The  fore-wheel  of  a  carriage  makes  4  revolutions 
more  than  the  hind-wheel  in  going  96  feet ;  but  if  the  cir- 
cumference of  the  fore-wheel  were  f  as  great,  and  of  the 
hind-wheel  ^  as  great,  the  fore-wheel  would  make  only  2 
revolutions  more  than  the  hind-wheel  in  going  the  same  dis- 
tance.    Find  the  circumference  of  each  wheel. 

61.  A  and  B  together  can  do  a  piece  of  work  in  4|^  days  ; 
but  if  A  had  worked  one-half  as  fast,  and  B  twice  as  fast, 
they  would  have  finished  it  in  4-^^  days.  In  how  many  days 
could  each  alone  perform  the  work? 

62.  A  and  B  run  a  race  of  300  yards.  The  first  heat,  A 
gives  B  a  start  of  40  yards,  and  beats  him  by  2  seconds ;  the 
second  heat,  A  gives  B  a  start  of  16  seconds,  and  is  beaten 
by  36  yards.     How  many  yards  can  each  run  in  a  second? 


158  ALGEBRA. 


XVII.  INVOLUTION. 

191.  Involution  is  the  process  of  raising  a  quantity  to 
any  required  power. 

This  is  effected,  as  is  evident  from  Art.  13,  by  taking  the 
quantity  as  a  factor  a  number  of  times  equal  to  the  exponent 
of  the  required  power. 

192.  If  the  quantity  to  be  involved  is  positive,  all  its 
powers  will  evidently  be  positive  ;  but  if  it  is  negative,  all 
its  even  powers  will  be  positive,  and  all  its  odd  powers  nega- 
tive.    Thus, 

(-a)2  =  (-a)x(-a)  =  +  a2 

(_a)3  =  (-a)x(-a)x(-a)  ^-a? 

(_a)^  =  (— a)  X  (—a)  X  (— «)  X  (— a)  =  +  a'*;  etc. 

Hence,  the  even  powers  of  any  quantity  are  positive;  and 
the  ODD  powers  of  a  quantity  have  the  same  sign  as  the  quan- 
tity itself. 

INVOLUTION  OF  MONOMIALS. 

193.  1.  Find  the  value  of  {ba^y, 

(5  d'cY  =  5  a^c  X  5  a^c  X  5  a'c  x  5  a^c  =  625  aV,  Ans, 

2.    Find  the  value  of  (-3m^)3. 

(-3m*)3  =  (-3m*)  X  (-3m4)  x  (-3m^)  = -27m^S  Ans, 

From  the  above  examples  we  derive  the  following  rule : 

Raise  the  numerical  coefficient  to  the  required  power ^  and 
multiply  the  exponent  of  each  letter  by  the  exponent  of  the 
required  power. 

Give  to  every  even  power  the  positive  sign,  and  to  every  odd 
power  the  sign  of  the  quantity  itself. 


mvoLUTioisr.  159 

EXAMPLES. 
Write  by  inspection  the  values  of  the  following : 

3.  (-a6V)*.  7.   (-6V)^  11.   (3a'6*c)«. 

4.  (-ba^by.  8.   (a^ftV)".  12.   (-Gx^'Y. 
6.   (x^y)'^.                9.   (-Sm^n)*.         13.   (4a'"62n)5. 

6.   (2mnV)^  10.   (4(1263^4)3.  14,   (-ja^y^i^)*. 

A  fraction  is  raised  to  any  power  by  raising  both  numera- 
tor  and  denominator  to  the  required  power, 

^  ,  /     2a;'"V     16a;*^ 

For  example,  (  —  -— • )  =  — — -• 

Write  by  inspection  the  values  of  the  following : 


"•  ©•• 

"■  (-^" 

.,.(- 

Ixyy 
3nJ 

'»■  (If)" 

IS.  (!«)•. 

»(- 

bcx'^Y 
4aV* 

SQUARE  OF  A  POLYNOMIAL. 

194.   We  find  by  multiplication : 

a  +  b  +  c 

a  +  b  +  G 

a^-\-    ab+    ac 

+    ab  +52^    5c 

+    ac         +    6c  +  c2 

a^  +  2ab-{-2ac-{-b^-\-2bc  +  c^ 

This  result,  for  convenience  of  enunciation,  may  be  written 
as  follows : 

(a  +  6  +  c)2  =  a«  +  6*  +  c^  H-  2a5  +  2ac  +  26c. 


160  ALGEBRA. 

In  a  similar  manner,  we  find  *. 

■i-2ab-h2ac  +  2ad-\-2bc-\-2bd-\-2cd', 
and  so  on. 

We  have  then  the  following  rule  for  the  square  of  any 
polynomial : 

Write  the  square  of  each  term,  together  with  twice  its 
product  by  each  of  the  following  terms. 

EXAMPLES. 

1.    Square  2 0^  —  3 «  — 5. 

The  squares  of  the  terms  are  4  a;'',  9a^,  and  25.  Twice 
the  first  term  into  each  of  the  following  terms  gives  the 
results,  —12ix^  and  —  20a^ ;  and  twice  the  second  term  into 
the  following  term  gives  the  result,  SOx.     Hence, 

{2a^-3x-5y=:ix^-\-di^  +  26-12x^-20a^  +  S0x 
=  4:X^-12x^-nx'  +  30x-]-25,  Ans, 

Square  the  following  expressions  : 

2.  a-b-\-c.  11.  x^-2x-\-5. 

3.  a  +  b-c.  ^  ^^'  2a^  +  3a^  +  l. 

4.  2ic2-fa;  +  l.                         13.  3a2-2a6-562. 
6.    a^-3a;4-l.                         14.  4m^ ■i-mn^ -Sn\ 

6.  a^+4a;-2.  15.  a  —  b-c-^d. 

7.  2x^-x-S.  le,  a-b  +  c-d. 

8.  3a2-5a  +  4.  17.  l+aj  +  a^  +  a^. 

9.  2a^  +  5a;-7.  18.  3x^-2x^-x-{-4:. 
10.  x  +  2y-Sz.  19.  x^-Ax'-2x-3. 


INVOLUTION.  161 


CUBE  OF  A  BINOMIAL. 
19S   We  find  by  multiplication  : 

{a  +  by=a'-\-2ab  +b' 
a  -i-b 


a^-{-2a'b+    ab"- 

a'b-{-2ab^-j-b^ 


(a  -hby  =  a^  -^-Sa'b  -\-3ab^  ■+■ 

{a-by  =  o?-2ab  +&' 
a  —b 


-    a'b  +  2ab''-b^ 


(a  _  5)3  =  a^  -^a'b  +  3ab^-  W 
That  is, 

The  cube  of  the  sum  of  two  quantities  is  equal  to  the  cube 
of  the  first,  plus  three  times  the  square  of  the  first  times  the 
second,  plus  three  times  the  first  times  the  square  of  the  sec- 
ond, plus  the  cube  of  the  second. 

The  cube  of  the  difference  of  two  quantities  is  equal  to  the 
cube  of  the  first,  minus  three  times  the  square  of  the  first  times 
the  second,  plus  three  times  the  first  times  the  square  of  the 
second,  minus  the  cube  of  the  second. 

EXAMPLES. 

1.  Find  the  cube  of  a-\-2b. 

(a  +  26)«  =  a3-f  3a2(26)  +  3a(26)2  +  (26)3 
=  a»  +  6a26  +  12a62  +  86^  Ans. 

2.  Find  the  cube  of  2x  —  3tf. 
{2x-3yy=^{2xy-3{2xy{3f)  +  ^2x){3fY-{Zyy 

=  8a^-36afy'  +  54.xy*-27f,  Ans. 


162  ALGEBRA. 

Find  the  cubes  of  the  following : 

3.  x-hS.  7.    3m2-l.  11.  2a^-3x. 

4.  2a;- 1.  8.   a^  +  4.  12.  Qx^-j-xij. 

5.  ab-cd.              9.    a +  56.  13.  3m  +  5n. 
_  6.    a  +  46.             10.    2x-6y.  14.  3a;2/-4a2. 

The  cube  of  a  trinomial  may  be  found  by  the  above 
method,  if  two  of  its  terms  be  enclosed  in  a  parenthesis  and 
regarded  as  a  single  term. 

16.    Find  the  cube  ot  x^  —  2x  —  l, 

(x'-2x-iy  =  l(a^-2x)-l2^ 

=  (x^-2xy-3{i>f-2xy  +  S(x'-2x)-l 

=  a^-6af  +  12a^-8x^-S(x^-4:i^-^4:X^) 

+  3(a^-2a?)-l 
=  a^  — 6a^+9i»^  +  4a^-9i^_(3^_l^  j^^g^ 

Find  the  cubes  of  the  following : 

16.  a^-x—1.       18.    a  +  b  —  c.  20.    x^  +  s^;.^!. 

17.  a-b  +  1.        19.    x'-2x-\-2.       21.    2ar^-3aj-l. 

ANY  POWER  OF  A  BINOMIAL. 
196.   By  actual  multiplication,  we  obtain  : 
(a-hby^a^+2ab  +  6^ 
(a-hby^a^  +  Sa'b  +  S  ab^  +  b^ 
(a-\-by=a'  +  4:a^b  +  6 a'b^  +  Aab^  +  b*]  etc. 

(a-'by=a^-2ab  +b^ 

(a-by  =  a^-Sa^b  +  3ab^-b^ 

(a  -  by  =  a'-4.a'b  +  6 a'b'  ^iab'  +  b']  etc. 


INVOLUTION.  163 

In  these  results  we  observe  the  following  laws : 

I.  The  number  of  terms  is  one  more  than  the  exponent 
of  the  binomial. 

II.  The  exponent  of  a  in  the  first  term  is  the  same  as  the 
exponent  of  the  binomial,  and  decreases  by  1  in  each  suc- 
ceeding term. 

III.  The  exponent  of  b  in  the  second  term  is  1,  and 
increases  by  1  in  each  succeeding  term. 

IV.  The  coeflficient  of  the  first  term  is  1  ;  and  of  the  sec- 
ond term,  is  the  exponent  of  the  binomial. 

V.  If  the  coeflBcient  of  any  term,  be  multiplied  by  the 
exponent  of  a  in  that  term,  and  the  result  divided  by  the 
exponent  of  b  increased  by  1 ,  the  quotient  will  be  the  coeflS- 
cient  of  the  next  term. 

VI.  If  the  second  term  of  the  binomial  is  negative,  the 
terms  in  the  result  are  alternately  positive  and  negative. 

By  aid  of  the  above  laws,  any  power  of  a  binomial  may 
be  written  by  inspection. 

EXAMPLES. 

1.   Expand  (a  +  xy. 

The  exponent  of  a  in  the  first  term  is  5,  and  decreases  by 
1  in  each  succeeding  term. 

The  exponent  of  x  in  the  second  tenn  is  1 ,  and  increases 
by  1  in  each  succeeding  term. 

The  coefficient  of  the  first  term  is  1  ;  of  the  second  term, 
5  ;  multiplying  the  coefficient  of  the  second  term  by  4,  the 
exponent  of  a  in  that  term,  and  dividing  the  result  by  the 
exponent  of  x  increased  by  1,  or  2,  we  have  10  for  the 
coefficient  of  the  third  term  ;  and  so  on.     Hence, 

(a  +  xy  =  a^-\-5a*x-\- 10 aV  +  lOaV -\-5ax*-\- x',  Ans, 


164  ALGEBRA. 

Note.  The  coefficients  of  terms  equally  distant  from  the  beginning 
and  end  of  the  expansion  are  equal.  Thus  the  coefficients  of  the  lat- 
ter half  of  an  expansion  may  be  written  out  from  the  first  half. 

2.    Expand  (1  -  xy. 

(1  -  a;)«  =  1^  -  6  .  1^ .  a;  -f  15  .  1* .  aj2  -  20  .  P .  a^ 

=  1  -6x-{-15x'-20af  +  16x'^-6x^-\-x^,  A71S. 

Note.  If  the  first  term  of  the  binomial  is  numerical,  it  is  con- 
venient to  write  the  exponents  at  first  without  reduction.  The  result 
should  afterwards  be  reduced  to  its  simplest  form. 

Expand  the  following : 

3.  (a -by.  7.  (1-0^)^  11.  {x-^y. 

4.  (a  +  by.  8.  (x-^yy.  12.  (a-Sy. 

5.  (a -by.  9.  {m-ny.  13.  (a  +  2)^ 

6.  {x-iy.        10.  (2-\-xy.        14.  (x-2y. 

16.    Expand  {Sm-ny. 

=  (3m)*-4(3m)3(n2)  +  6(3m)2(n2)2 

-A{3m)(n'y-\-(7i'y 
=  81  m*  —  108  m^n^  +  54  mV  — 12  mn^  +  n^  Ans. 

Note.   If  either  term  of  the  binomial  has  a  coefficient  or  exponent 
other  than  unity,  it  should  be  enclosed  in  a  parenthesis  before  apply- 
ing the  laws- 
Expand  the  following : 

16.  (a-3xy.        18.   (a'  +  bcy.         20.   (2a'-{-by. 

17.  (3  +  26)^         19.   (a;3-4)^  21.   {2m^-Sny. 


EVOLUTION.  165 


XVIII.  EVOLUTION. 

197.  If  a  quantity  be  resolved  into  any  number  of  equal 
factors,  one  of  these  factors  is  called  a  Boot  of  the  quantity. 

198.  Evolution  is   the   process  of   finding   any  required 
root  of  a  quantity.     This  is  effected,  as  is  evident  from  the 
preceding  article,  by  finding  a  quantity  which,  when  raised  )i 
to  the  proposed  power,  will  produce  the  given  quantit3\  '  / 

199.  The  Radical  Sign,  ^,  when  prefixed  to  a  quantity, 
indicates  that  some  root  of  the  quantity  is  to  be  found. 

Thus,  -y/a  indicates  the  second  or  square  root  of  a ; 
■^a  indicates  the  third  or  cube  root  of  a ; 
■^a  indicates  ihe  fourth  root  of  a  ;  and  so  on. 

The  ijidex  of  the  root  is  the  figure  written  over  the  radical 
sign.  When  no  index  is  written,  the  square  root  is  under- 
stood. 

EVOLUTION  OF  MONOMIALS. 

200.  Required  thfe  cube  root  of  a^b^c^. 

By  Art.  198,  we  are  to  find  a  quantity  which,  when  raised 
to  the  third  power,  will  produce  a^6V.  That  quantity  is 
evidently  a6V.     Hence, 

■V^iFW^  =  ab'(^. 

That  is,  any  root  of  a  monomial  is  obtained  by  dividing  the 
exponent  of  each  factor  by  the  index  of  the  required  root. 

201.  from  the  relation  of  a  root  to  its  corresponding 
power,  it  follows  from  Art.  192  that : 

1.  The  odd  roots  of  a  quantity  have  the  same  sign  as  ihe 
quantity  itself. 

Thus,  Va*  =  a,  and  ■{/  —  a^  =  —  a. 


166  ALGEBRA. 

2.  The  even  roots  of  a  positive  quantity  are  either  positive 
or  negative. 

For  the  even  powers  of  either  a  positive  or  a  negative 
quantity  are  positive. 

Thus,  -y/a"^  =  a  or  —  a  ;  that  is,  -^a^  =±a. 

Note.  The  sign  ±,  called  the  double  sign,  is  prefixed  to  a  quantity 
when  we  wish  to  indicate  that  it  is  either  +  or  — . 

3.  The  even  roots  of  a  negative  quantity  are  impossible. 

For  no  quantity  when  raised  to  an  even  power  can  pro- 
duce a  negative  result.  Such  roots  are  called  imaginary 
quantities. 

202.  From  Arts.  200  and  201  we  derive  the  following 
rule : 

Extract  the  required  root  of  the  numerical  coefficient,  and 
divide  the  exponent  of  each  letter  by  the  index  of  the  7'oot. 

Give  to  every  even  root  of  a  positive  quantity  the  sign  ± , 
and  to  every  odd  root  of  any  quantity  the  sign  of  the  quantity 
itself. 

Note.  Any  root  of  a  fraction  may  be  found  by  taking  the  required 
root  of  each  of  its  terms. 

EXAMPLES. 

1.    Find  the  square  root  of  9a^6V. 


By  the  rule,      V9  a*6V  =  ±  3  a^^c^  Ans. 
2.    Find  the  fifth  root  of  —  32  aV'". 


V-32a^V'"  =  -2a2a;'",  Ans. 
Find  the  values  of  the  following  : 


3. 

■^-125aTy. 

7. 

8. 

9. 

10. 

■</-8a«6V. 

11. 
12. 
13. 
14. 

-\/81  n^V. 

4. 

V49«^&2c^. 

Vl21a^c2. 

S/-243c^"(^^^. 

5. 

-v/m^7iy«. 

Va'""&'"^. 

■\/64.a''b''c\ 

6. 

y^Ua'b', 

VSla'^'x"^'. 

^^n-^Sy^-a^ 

15 


EVOLUTION,  167 

/9^  --      5l~~32^  10      3^    64 mV 

le    ^1^^^'  18     'IIZI         20    'F^lL 

SQUARE  ROOT  OF  POLYNOMIALS. 

203.  Since  (a  + 6)^  =  a^  + 2afe +  6^  we  know  that  the 
square  root  of  a^  +  2  a5  +  6Ms  a  +  6. 

It  is  required  to  find  a  process  by  which,  when  the  quan- 
tity a^ -{- 2  ab -\- b^  is  given,  its  square  root,  a +  5,  may  be 
determined. 


a2  +  2a6H-62 

/,2 


2a-\-b 


^   I   ^       The  square  root  of  the  first  term  is  a, 
which  is  the  first  term  of  the  root.     Sub- 
tracting its   square  from  the  given  ex- 
2  a6  +  6  pression,  the  remainder  is   2ab  +  i',   or 

2ab-\-b^  {2a-\-b)b.      Dividing   the   first  term  of 

this  remainder  by  2  a,  or  twice  the  first 
term  of  the  root,  we  obtain  b,  the  second  term.  This  being  added  to 
2  a,  gives  the  complete  divisor  2  a  +  6 ;  which,  when  multiplied  by  b, 
and  the  product,  2  a6  +  P,  subtracted  from  the  remainder,  completes 
the  operation. 

From  the  above  process  we  derive  the  following  rule : 

Arrange  the  terms  according  to  the  2)owers  of  some  letter. 

Find  the  square  root  of  the  first  term,  write  it  as  the  first 
term  of  the  root,  and  subtract  its  square  from  the  given 
expression. 

Divide  the  first  term  of  the  remainder  by  twice  the  first  term 
of  the  root,  and  add  the  quotient  to  the  root  and  also  to  the 
divisor. 

Multiply  the  complete  divisor  by  the  term  of  the  root  last 
obtained,  arid  subtract  the  product  from  the  remainder. 

If  other  terms  remain,  proceed  as  before,  doubling  tho 
part  of  the  root  akeady  found  for  the  next  trial-divisor. 


168 


ALGEBRA. 


EXAMPLES. 

204.    1 .  Find  the  square  root  of  9x'^  —  30  aV  +  25  a^. 

9a;*-30aV  +  25a«    3a^-5a\  Ans. 
9a^ 


6a^ 


oa"" 


-30a3ar^  +  25a« 
-30aV  +  25a« 


The  square  root  of  the  first  term  is  3  x^,  which  is  the  first  term  of  the 
root.  Subtracting  9x*  from  the  given  expression,  we  have  —  SOa^x^  as 
the  first  term  of  the  remainder.  Dividing  this  by  twice  the  first  term 
of  the  root,  6  x^,  we  obtain  the  second  term  of  the  root,  —  6  a^,  which, 
added  to  6  x'^,  completes  the  divisor  6  a:^  —  5  a^.  Multiplying  this  divi- 
sor by  —  Sa^,  and  subtracting  the  product  from  the  remainder,  there  is 
no  remainder.     Hence,  Sx"^  —  5a^  is  the  required  square  root. 


2.    Find  the  square  root  of 

Arranging  according  to  the  descending  powers  of  x, 


6ar^+2a;2 


3a.'3-f2a^-2a;-l, 
Ans. 


12ic^ 
12a^+    Ax^ 


6a^-{-4:Xr-2x 


-12a;* 

-12a^-8x'-\-4:X^ 


6a^  +  4ar^-4£c-l 


-6a^-4x-2  4-4a?+l 


It  will  be  observed  that  each  trial-divisor  is  equal  to  the 
preceding  complete  divisor,  with  its  last  term  doubled. 

Note.  Since  every  square  root  has  the  double  sign  (Art.  201),  the 
result  may  be  written  in  a  different  form  by  changing  the  sign  of  each 
term.     Thus,  in  Example  2,  another  form  of  the  answer  is 

-3x3-2ar2  +  2x+l. 


EVOLUTION.  169 

Find  the  square  roots  of  the  following : 
3.    a'-ia^  +  Qa'-Aa  +  l. 

6.  9-12a:  +  10a^-4ar^  +  a^. 

7.  4:0x-\-25-Ux--\-9x'-24a^. 

8.  m2-f  2m- !--  +  —• 

m      m^ 

9.  4:a*  +  Ub^-20cv'b-S0ab^-\-o7a^b^. 
-    10.    28a^4-4iB^-14a;+l4-45a^. 

11.  a'-{-b'-{-c'-2ab-2a€-\-2bc. 

12.  iB^.^  4^/2 _^  9^2  _4^.y_^g^.^_122/2;. 

13.  9aj«  +  30ar'  +  25a;^-42ar^-70if2  +  49. 

14.  16c«-40c4-24c3+25c2  +  30c  +  9. 

15.  9  +  a'-  +  30a  -  4cf  +  13a2  +  14a^  -  14ft=^. 

16.  4x^  —  4:X^y  —  3xY—6a^f-{-i)x^y^-^4:xy^-\-4y^, 

/l1.  25x^-Uo^-4:Ox-\-4x^-\-25-i-4:6x^-12x^, 

iQ     a*      2a^b  ,  4cr6-        .3  .    b* 
18. ^ a¥  H 

9  3  3  4 

19.  da^-VIa^y-hlOxY-iea^y^-j-dx^y'-^xy^-^iy^. 

Find  to  four  terms  the  approximate  square  roots  of  the 
following : 

20.  1  +  x.  22.  a^~iab-\-bK 


21.  l-2a.  23.  4x^-^2y. 


170  ^  ALGEBRA. 


SQUARE  ROOT  OF  NUMBERS. 

205.  The  method  of  Art.  204  may  be  used  to  extract  the 
square  roots  of  arithmetical  numbers. 

The  square  root  of  100  is  10;  of  10,000  is  100;  etc. 
Hence,  the  square  root  of  a  number  less  than  100  is  less 
than  10  ;  the  square  root  of  a  number  between  10,000  and 
100  is  between  100  and  10  ;  and  so  on. 

That  is,  the  integral  part  of  the  square  root  of  a  number  of 
one  or  two  figures,  contains  one  figure  ;  of  a  number  of  three 
or  four  figures,  contains  two  figures  ;   and  so  on.     Hence, 

If  a  point  he  placed  over  every  second  figure  in  any  integral 
number^  beginning  with  the  units'  place ^  the  number  of  points 
shows  the  number  of  figures  in  the  integral  part  of  its  square 
root. 

206.  Let  it  be  required  to  find  the  square  root  of  4624. 


4624 
a2  =  3600 


120  +  8 
=  2a4-6 


60+8       Pointing  the  number  according  to 


=  a-\-b 


the  rule  of  Art.  205,  we  see  that  there 
are  two  figures  in  the  integral  part  of 
1024  the  square  root. 

1024  Let  a  denote  the  value  of  the  num- 

ber  in  the  tens'  place  in  the  root,  and 
b  the  number  in  the  units'  place.  Then  a  must  be  the  greatest  multi- 
ple of  10  whose  square  is  less  than  4624 ;  this  we  find  to  be  60.  Sub- 
tracting a2,  that  is,  the  square  of  60  or  8600,  from  the  given  number, 
the  remainder  is  1024.  Dividing  the  remainder  by  2  a  or  120,  we  have 
8  as  the  value  of  b.  Adding  this  to  120,  multiplying  the  result  by  8, 
and  subtracting  the  product,  1024,  there  is  no  remainder.  Hence, 
60  +  8  or  68  is  the  required  square  root. 

The  ciphers  being  omitted  for  the  sake  of  brevity,  the  work 
will  stand  as  follows  : 


4624 
36 


68 


128  I  1024 
I  1024 


EVOLUTION.  171 

From  the  above  process  we  derive  the  following  rule : 

Separate  the  number  into  periods  by  pointing  every  second 
.figure,  beginning  with  the  units'  pla^e. 

Find  the  greatest  square  in  the  left-hand  period,  and  write 
its  square  root  as  the  first  figure  of  the  root;  subtract  its 
square  from  the  mimber,  and  to  the  residt  bring  down  the  next 
period. 

Divide  this  remainder,  omitting  the  last  figure,  by  twice  the 
part  of  the  root  already  found,-  and  annex  the  quotient  to  the 
root  and  also  to  the  divisor. 

Multiply  the  complete  divisor  by  the  figure  of  the  root  last 
obtained,  and  subtract  the  x>roduct  from  the  remainder. 

If  other  periods  remain,  proceed  as  before,  doubling  the 
part  of  the  root  already  found  for  the  next  trial-divisor. 

Note  1.  It  should  be  observed  that  decimals  require  to  be  pointed 
to  the  right. 

Note  2.  As  the  trial-divisor  is  an  incomplete  divisor,  it  is  sometimes 
found  that  after  completion  it  gives  a  product  greater  than  the  remain- 
der. In  such  a  case,  the  last  root-figure  is  too  large,  and  one  less  must 
be  substituted  for  it. 

Note  3.  If  any  root-figure  is  0,  annex  0  to  the  trial-divisor,  and 
bring  down  to  the  remainder  the  next  period. 

EXAMPLES. 
207.    1.  Find  the  square  root  of  49.449024. 


49.449024 

7.032,  Ans, 

49 

1403 

4490 
4209 

14062 

28124 
28124 

Since  the  second  root-figure  is  0,  we  annex  0  to  the  trial- 
divisor  14,  and  bring  down  to  the  remainder  the  next  period, 
90. 


172 


ALGEBRA. 


Extract  the  square  roots  of  the  following : 

2.  45796.  6.    .247009.  10.    446.0544. 

3.  273529.       7.  .081796.     11.  .0022448644. 

4.  654481.       8.  .521284.     12.  811440.64. 

5.  33.1776.      9.  1.170724.     13.  .68112009. 


If  there  is  a  final  remainder,  the  given  number  has  no 
exact  square  root ;  but  we  may  continue  the  operation  by 
annexing  periods  of  ciphers,  and  thus  obtain  an  approximate 
value  of  the  square  root,  correct  to  any  desired  number  of 
decimal  places. 

14.    Extract  the  square  root  of  12  to  five  figures. 

3.4641...,  A71S. 


12.06060606 

9 

64 

300 
256 

eSG 

4400 
4116 

6924 

I:  28400 
27696 

69281 

70400 
69281 

1119 

Extract  the  square  roots  of  the  following  to  five  figures : 

15.  2.  18.    11.  21.    .7.  24.  .001. 

16.  3.  19.   31.  22.    .08.  25.  .00625. 

17.  5.  20.    17.3.  23.    .144.  26.  2.08627. 

The  square  root  of  a  fraction  may  be  obtained  by  taking 
the  square  roots  of  its  terms. 


EVOLUTio:Nr.  173 

If  the  denominator  is  not  a  perfect  square,  it  is  better  to 
reduce  the  fraction  to  an  equivalent  fraction  whose  denomi- 
nator is  a  perfect  square. 

3 
Thus,  to  obtain  the  square  root  of  -,  we  should  proceed 

as  follows : 

J5  =  JA=V6  =  2,M?M^  =  . 61237,.. 
\8      \16        4  4 

Extract  the  square  roots  of  the  following  to  five  figures : 


27.  1 

4 

"f- 

-1 

33. 

11 

8* 

35.   ^. 

18 

^■a- 

»i. 

-M 

34. 

9 
10* 

72 

CUBE  ROOT  OF  POLYNOMIALS. 

208.  Since  (a  4-  &) ^  =  a'  +  3  a^^  +  3  aW  +  6■^  we  know  that 
the  cube  root  of  a^  -f  3  a^h  +  3  cib'^  -f-  h^  is  a  +  6. 

It  is  required  to  find  a  process  by  which,  when  the  quan- 
tity a^  -f  3  a^6  +  3  ab^  -f-  h^  is  given,  its  cube  root,  a  +  6,  may 
be  determined. 

a-^b 


3a2  +  3a6-f-62 

^a'h  +  Zah^  +  h^ 

The  cube  root  of  the  first  term  is  a,  which-  is  the  first  term  of  the 
root.  Subtracting  its  cube  from  the  given  expression,  the  remainder 
is  ^a}h  +  3a&2  +  h^,  or  (Sa^  +  3a6  +  6^)  h.  Dividing  the  first  term  of 
this  remainder  by  3  a^,  or  three  times  the  square  of  the  first  term  of 
the  root,  we  obtain  h,  the  second  term. 

Adding  to  the  trial-divisor  3  aft,  that  is,  three  times  the  product  of 
the  first  term  of  the  root  by  the  second,  and  6^,  that  is,  the  square  of  the 
last  term  of  the  root,  completes  the  divisor,  Sa^  +  3a6  +  h"^.  This  being 
multiplied  by  h,  and  the  product,  Za% -{-Zab'^  ■\- h^,  subtracted  from 
the  remainder,  completes  the  operation. 


174  ALGEBRA. 

From  the  above  process  we  derive  the  following  rule : 

Arrange  the  terms  according  to  the  powers  of  some  letter. 

Find  the  cube  root  of  the  first' term ^  write  it  as  the  first  term 
of  the  root,  and  subtract  its  cube  from  the  given  expression. 

Divide  the  first  term  of  the  remainder  by  three  times  the 
square  of  the  first  term  of  the  root,  and  write  the  quotient  as 
the  next  term  of  the  root. 

Add  to  the  trial-divisor  three  times  the  product  of  the  first 
term  of  the  root  by  the  second,  and  the  square  of  the  second 
term. 

Multiply  the  complete  divisor  by  the  term  of  the  root  last 
obtained,  and  subtract  the  product  from  the  remainder. 

If  other  terms  remain,  proceed  as  before,  taking  three 
times  the  square  of  the  root  already  found  for  the  next  trial- 
divisor. 

EXAMPLES. 

209.  1.  Find  the  cube  root  of  S:(^ -2>Qd^y +  64.x^y^  — 
272/«. 


8a^ 


12a^-18a^2/+92/' 


2  a^  —  3  2/,  Ans. 


-  36 x*y  +  54:x'y^- 27  f 
-S6x'y  +  54taPy^-27f 


The  cube  root  of  the  first  term  is  2  x"^,  which  is  the  first  term  of  the 
root.  Subtracting  Sx^  from  the  given  expression,  we  have  —S6x*i/  as 
the  first  term  of  the  remainder.  Dividing  this  by  three  times  the 
square  of  the  first  term  of  the  root,  12a:*,  we  obtain  —  3y  as  the  second 
term  of  the  root.  Adding  to  the  trial-divisor  three  times  the  product 
of  the  first  term  of  the  root  by  the  second,  — 18  x'^y,  and  the  square  of 
the  second  term,  9y^,  completes  the  divisor,  12  x*  —  lSx^y-{-9y^. 
Multiplying  this  by  —Sy,  and  subtracting  the  product  from  the  re- 
mainder, there  is  no  remainder.  Hence,  2^2  —  3y  is  the  required  cube 
root. 

2.   Find  the  cube  root  of  40a^-6««- 96a;  +  aJ«- 64. 


EVOLUTION. 


175 


An-anging  according  to  the  descending  powers  of  ic, 


o^^2x~4:. 


Ans. 


3a^_6a^4-4a^ 


3aj^-12a;3+12a^ 


3x*-12a^ 


+  24a;+16 


-12a;^+48a^-96a;-64 
-12x*+ASa^-dQx-64: 


The  second  complete  divisor  is  formed  as  follows : 

The  trial-divisor  is  3  times  the  square  of  the  root  already  found ; 
that  is,  3  {x^  -2x)\  or  3a:*  -  12^3  +  12x2.  Three  times  the  product  of 
the  root  already  found  by  the  last  term  of  the  root  is  3(— 4)(x'  —  2a:), 
or  —  12a:2  +  24  a:,*  and  the  square  of  the  last  root-term  is  16.  Adding 
these,  we  have  for  the  complete  divisor  3x*  —  12  a:'  -f-  24x  +  16. 

Find  the  cube  roots  of  the  following : 

3.  i-Qy-^i2y'-8f. 

4.  27iB«H-27a;*-f  9ar+l. 

6.  54:xf-^27f+S6x'y-\-8a^. 

6.  64a»-  lUa'xy-{- 108 ax'y^ -  27 a? f, 

7.  a36-f-6a^-40a^-f  96a;-64. 

8.  f-1^5f-Sy'-Sy. 

9.  15x*-6x-6a^-\-15x^-{-l-\-a^-20a?. 

10.  9a^-21ic2-36ar'-|-8a^-9a;+42a^-l. 

11.  8a«-12ci«-54a*  +  59a3  4-135a2-75a-125. 

12.  30ic2-12a^-12a;-f8-25a^+8a;«  +  30aj*. 


13.  xf^  +  Sxi'y-3xY-na^f-{-6a^y*-\-12xf-8f. 

14.  27 a«  -  54  a'^ft  4-  9  a%^  -f  28  a^b^  -  3  a%^  -  6  aft*  -  6«. 


176  ALGEBRA. 


CUBE  ROOT  OF  NUMBERS. 

210.  The  method  of  Art.  209  may  be  used  to  extract  the 
cube  roots  of  arithmetical  numbers. 

The  cube  root  of  1000  is  10;  of  1,000,000  is  100;  etc. 
Hence,  the  cube  root  of  a  number  less  than  1000  is  less  than 
10  ;  the  cube  root  of  a  number  between  1,000,000  and  1000 
is  between  100  and  10  ;  and  so  on. 

That  is,  the  integral  part  of  the  cube  root  of  a  number  of 
one,  two,  or  three  figures,  contains  one  figure  ;  of  a  number 
of  four,  five,  or  six  figures,  contains  two  figures ;  and  so  on. 
Hence, 

If  a  point  he  placed  over  every  third  figure  in  any  integral 
number^  beginning  with  the  units'  place ^  the  number  of  points 
shows  the  number  of  figures  in  the  integral  part  of  its  cube 
root. 

211.  Let  it  be  required  to  find  the  cube  root  of  157464. 

50  -f-  4       Pointing    the    number    according 

there  are  two  figures  in  the  integral 
part  of  the  cube  root. 

Let  a  denote  the  value  of  the 
number  in  the  tens'  place  in  the 
root,  and  h  the  number  in  the  units' 
place.  Then  a  must  be  the  greatest 
multiple  of  10  whose  cube  is  less  than  157464 ;  this  -we  find  to  be  50. 
Subtracting  a^,  that  is,  the  cube  of  50  or  125000,  from  the  given  num- 
ber, the  remainder  is  32464.  Dividing  this  remainder  by  Sa^,  that  is, 
3  times  the  square  of  50  or  7500,  we  obtain  4  as  the  value  of  h.  Adding 
to  the  trial-divisor  3  ah,  that  is,  3  times  the  product  of  50  and  4,  or  600, 
and  b^,  or  16,  we  have  the  complete  divisor  8116.  Multiplying  this  by 
4,  and  subtracting  the  product,  32464,  there  is  no  remainder.  Hence, 
50  +  4  or  54  is  the  required  cube  root. 

The  ciphers  being  omitted  for  the  sake  of  brevity,  the  work 
will  stand  as  follows  : 


L57464  1 

a3=  125000 

3a2  =7500 

32464 

3a6=  600 

&^=   16 

8116 

32464 

EVOLUTION. 

157464 

54 

125 

7500 

32464 

600 

16 

8116 

32464 

177 


From  the  above  process,  we  derive  the  following  rule : 

Separate  the  number  into  periods  by  pointing  every  third 
figure^  beginning  with  the  units'  place. 

Find  the  greatest  cube  in  the  left-hand  period^  and  write  its 
cube  root  as  the  first  figure  of  the  root;  subtract  its  cube  from 
the  number^  and  to  the  result  bring  down  the  next  period. 

Divide  this  remainder  by  three  times  the  square  of  the  root 
already  founds  with  two  ciphers  annexed^  and  write  the  quotient 
as  the  next  figure  of  the  root. 

Add  to  the  trial-divisor  three  times  the  product  of  the  last 
root  figure  and  the  part  of  the  root  previously  found,  with  one 
cipher  annexed^  and  the  square  of  the  last  root  figure. 

Multiply  the  complete  divisor  by  the  figure  of  the  root  last 
obtained,  and  subtract  the  product  from  the  remainder. 

If  other  periods  remain,  proceed  as  before,  taking  three 
times  the  square  of  the  root  already  found  for  the  next 
trial-divisor. 

The  notes  ta  Art.  206  apply  with  equal  force  to  examples 
in  cube  root,  except  that  in  Note  3  ttco  ciphers  should  be 
annexed  to  the  trial-divisor. 

212.  In  the  illustration  of  Art.  208,  if  there  had  been 
more  terms  in  the  given  quantity,  the  next  trial-divisor 
would  have  been  three  times  the  square  of  a  +  6 ;  that  is, 
3a^ -{-Qab-\-3b^.  We  observe  that  this  is  obtained  from  the 
preceding  complete  divisor,  3  a^  +  3  a6  +  6",  by  adding  to  it 
its  second  term,  3  ab,  and  twice  its  third  term,  25^    We  may 


178 


ALGEBRA. 


then  use  the  following  rule  for  forming  the  successive  trial- 
divisors  in  the  cube  root  of  numbers  : 

To  the  preceding  complete  divisor^  add  its  second  term  and 
twice  its  third  term;  and  annex  two  ciphers  to  the  result. 


EXAMPLES. 
213.    1.  Find  the  cube  root  of  8.144865728. 


8.144865728 

2.012,  Ans. 

8 

120000 

144865 

600 

1 

120601 

120601 

600 

24264728 

2 

12120300 

12060 

4 

24264728 

1213236 

4 

Since  the  second  root-figure  is  0,  we  annex  two  ciphers  to  the  trial- 
dirisor  1200,  and  bring  down  to  the  remainder  the  next  period,  865. 

The  second  trial-divisor  is  formed  by  the  rule  of  Art,  212.  The  pre- 
ceding complete  divisor  is  120601 ;  adding  its  second  term,  600,  and 
twice  its  third  term,  2,  we  have  121203;  annexing  two  ciphers  to  this,  we 
obtain  the  result  12120300. 


Extract  the  cube  roots  of  the  following  : 


2.  29791. 

3.  97.336. 

4.  .681472. 

5.  1860867. 

6.  1.481544, 


7.  .000941192. 

8.  8.242408. 

9.  51478848. 

10.  10077.696. 

11.  .517781627. 


12.  116.930169. 

13.  .031855013. 

14.  .724150792. 

15.  1039509.197. 

16.  .000152273304. 


EVOLUTION.  179 

Extract  the  cube  roots  of  the  following  to  four  figures  : 

17.  2.       ^Id.    7.2.  21.   |.      >23.   ^. 

18.  6.       -{^20.   .03.  22.  |.     -<  24.  |. 

214.  When  the  index  of  the  required  root  is  the  product 
of  two  or  more  numbers,  we  may  obtain  the  result  by  suc- 
cessive extractions  of  the  simpler  roots. 

For,  by  Art.  198,     ("^^/'a)""*  =  a. 

Taking  the  nth  root  of  both  members, 

('»^a)-=-;ya.  (1) 

Taking  the  mth  root  of  both  members  of  (1), 

-ya=  V(-v/a)- 
That  is, 

TJie  mnth  root  of  a  quantity  is  equal  to  the  mth  root  of 
the  nth  root  of  that  quantity. 

For  example,  the  fourth  root  is  the  square  root  of  the 
square  root ;  the  sixth  root  is  the  cube  root  of  the  square 
root;  etc. 

EXAMPLES. 

Find  the  fourth  roots  of  the  following : 

1.    16a;*-96a^?/H-216a^/-216a;2/3_f_gi2/4^ 

3.  a^-8a;^+16aj«+16ar'-56a;^-32a;s+64a;2^g4a,^lg^ 

Find  the  sixth  roots  of  the  following : 

4.  ai2-6ai«  +  15a«-20a«  +  15a^-6a2  4.i. 

0.    64a^-|-192««  +  240a^+160a;»  +  60ar'-f-12a;  +  l. 


180  ALGEBRA. 


XIX.    THE   THEORY    OP    EXPONENTS. 

215.  In  the  preceding  chapters  we  have  considered  an 
exponent  only  as  a  positive  whole  number.  It  is,  however, 
found  convenient  to  employ  fractional  and  negative  expo- 
nents ;  and  we  proceed  to  define  them,  and  to  prove  the 
rules  for  their  use. 

216.  In  Art.  13  we  defined  a  positive  integral  exponent  as 
indicating  how  many  times  a  quantity  was  taken  as  a  factor ; 
thus, 

a"*  signifies  axax«X  •••  torn  factors. 

"We  have  also  found  the  following  rules  to  hold  when  m 
and  n  are  positive  integers  : 

I.    a'^  X  a^  =  a''*+'*.  (Art.  79.) 

II.   ((X'«)"  =  a'"».  (Art.  193.) 

217.  The  definition  of  Art.  13  has  no  meaning  unless  the 
exponent  is  a  positive  integer,  and  we  must  therefore  adopt 
new  definitions  for  fractional  and  negative  exponents.  It 
is  convenient  to  have  all  forms  of  exponents  subject  to  the 
same  laws  in  regard  to  multiplication,  division,  etc.,  and  we 
shall  therefore  assume  Rule  I.  to  hold  for  all  values  of  m 
and  w,  and  find  what  meanings  must  be  attached  to  fractional 
and  negative  exponents  in  consequencco 

218.  Required  the  meaning  of  a^. 

Since  Rule  I.  is  to  hold  universally,  we  must  have 

a^  X  a^  X  a^  =  a^ "^^"^^  =  a^. 

That  is,  a^  is  such  a  quantity  that  when  raised  to  the  third 
power  the  result  is  a^.  Hence  (Art.  198),  a^  must  be  the 
cube  root  of  a^ ;  or,  a^  =  ^a^. 


THEORY  OF  EXPONENTS.  181 

We  will  now  consider  the  general  case : 

p 
Required  the  meaning  of  a?,  where  p  and  q  are  positive 

integers. 

p       p       p 
By  Rule  I.,  a^  x  a^  x  a^  X  •••  to  q  factors 

P 

That  is,  ai  is  such  a  quantity  that  when  raised  to  the  gth 

p 
power  the  result  is  a^.     Therefore  a"  must  be  the  gth  root  of 

9^'i  or, 

a^  =  -{/a^. 

Hence,  in  a  fractional  exponent,  the  numerator  denotes  a 
power  and  the  denominator  a  root. 

For  example,  a*  =  -y/a^ ;  c^  =  ^c^  ;  x^  =  -^x ;  etc. 

EXAMPLES. 
219.   Express  the  following  with  radical  signs  : 

1.  aK      3.  2ci  5.  x^y^.       7.  4a^5t.         9.  5yh^. 

2.  b^.       4.  3am*.      6.  m^ni      8.  2c^di        10.  ab^c^dK 

Express  the  following  with  fractional  exponents  : 

11.  ^x\  13.  V^-  15.  S^m\  17.  ^a*^&. 

12.  -^2/'-  14.  ^c.  16.  4-^a9.  18.  V^-v^^/'- 

19.  SVwi'a/w*-  20.  2a^x^y"'. 

The  value  of  a  numerical  quantity  affected  with  a  fractional 
exponent  may  be  found  by  first  extracting  the  root  indi- 
cated by  the  denominator,  and  then  raising  the  result  to  the 
power  indicated  by  the  numerator. 

Thus,  (-8)*  =  (-\Ar8)2  =  (-2)^=4. 


182  ALGEBRA. 

Find  the  values  of  the  following : 

21.  9i  23.  36i  25.   (-27)i         27.  64^ 

22.  27i  24.  lei  26.  (-32)i         28.  (-216)i 

220.  Required  the  meaning  of  a". 

Since  Rule  I.  is  to  hold  universally,  we  must  hare 

Therefore  aP  must  be  equal  to  1 . 

That  is,  any  quantity  whose  exponent  is  0  is  equal  to  1. 

221.  We  pass  next  to  the  case  of  negative  exponents. 
Required  the  meaning  of  a~^. 

By  Rule  I.,        a~^  x  a^  =  a-^+s  =  a«  =  1.  (Art.  220.) 

Hence,  a~^=—' 

a^ 

We  will  now  consider  the  general  case  : 

Required  the  meaning  of  a~'^^  n  being  integral  or  fractional. 

By  Rule  I.,       a""  x  a^  =  a-^+"  =  a^=  1. 

Hence,  a"^  =  — 

a^ 


For  example, 


1  _*!_,_! 

_;  a  ^  =  --;  Sx~y  2  _. 


,-2_A.  ^-^_  -  ..3a.-y?^   3   .  etc. 
»2  a^  xy^ 


222.  In  connection  with  Art.  221,  the  following  principle 
may  be  noticed : 

Any  factor  of  the  numerator  of  a  fraction  may  he  transferred 
to  the  denominator^  or  any  factor  of  the  denominator  to  the 
numerator,  if  the  sign  of  its  exponent  he  changed. 


THEORY  OF  EXPONENTS.  183 


Thus,  the  fraction  ^  can  be  written  in  any  of  the  forms 
cdr 


o'       0^^     _i etc 


EXAMPLES. 
223.  Write  the  following  with  positive  exponent*  i 

1.  a^y-\  5.   a-'b-\  9.    6a-'b-^c. 

2.  x-'y^.  6.   3ah-^.  10.   2m-«w-*. 

3.  m^rrK  7.    2x-*y~K  11.   ^x'^y'K 

4.  ^xy~K  8.   a-^6-V.  12.   a-26-^c"i 

Transfer  the  literal  factors  from  the  denominators  to  tha 
numerators  in  the  following : 

13.  1. 

X 

16.  A. 

Transfer  the  literal  factors  from  the  numerators  to  the 
denominators  in  the  following : 

oo    ^^  OK     2c"^  OQ       -a  1 

22.  -—•  25.   — - —  28.   m  ^w^. 

3  5 

23.  ^*.  26.   3ai  29.   ^. 

24    £!!  27    ^^~^^  30    i^^ll^. 

*     2  '  *      A*    '  *        Sc'     * 


16. 

1 

2a;* 

17. 

3c 

18. 

ad" 

19. 

5a2 
2  6c3 

20. 

2x^y^ 

21. 

^x 

184  ALGEBRA. 

224.  Since  the  definitions  of  fractional  and  negative  expo- 
nents were  obtained  on  the  supposition  that  Rule  I.,  Art. 
216,  was  to  hold  universally,  we  have  for  any  values  of  m 

and  w, 

«»«  X  a**    =  «"*+**. 

For  example, 


a^  X  a 

a  xa~^  =  a^~2-  =a,~^;  etc. 


EXAMPLES. 
Find  the  values  of  the  following : 


1. 

a^  X  a"^ 

6. 

SaXoTK 

11. 

2c"^x3a^c8. 

2. 

a?  X  a-\ 

7. 

5c-3x3c"i 

12. 

2a-364xa6~^ 

3. 

x-^  X  x-\ 

8. 

a'  X  -^a^. 

13. 

a^,-tx-/. 

4. 

n^  X  n~^. 

9. 

x-^  X  ^a;-^ 

14. 

^a;x  5^x-^ 

5. 

2x^  X  x~^. 

10. 

m^X   ,^  . 

15. 

1      X     ^ 

16.   Multiply  a  4-  2  a^  -  3  a*  by  2  -  4 a"^  -  6 a"^. 

a  +  2a^    -3a* 
2  — 4a~*  — 6a"^ 

2a  +  4a^    -6a* 
-4a^    -8a*  +  12 

-6a*-12  +  18a~* 


2a  -20a*  +18a  *,  ^ws. 


Note.  In  examples  like  the  above,  it  should  be  borne  in  mind  that 
any  quantity  whose  exponent  is  0  is  equal  to  1.     (Art,  220.) 


THEORY  OF  EXPONENTS.  185 

Multiply  the  following : 

17.  a2-24-ci~'by  a2+2  +  a~^. 

18.  a^  +  a^x^-^  x^  by  a^  —  a#. 

19.  x~^  —  x~^-\-x~^-lhyx~^-\-l. 

20.  a;-2-2a3Fi  +  l-2a;by  a;-3  +  2a;-2. 

21.  3a-i  -  a-26-i  +  a-^h'^  by  e>a?h^  +  2a^h  +  2a. 

22.  2a;^— 3a;*  — 4  +  a;~^by  3a;^  +  a;  — 2a;^. 

23.  x-^y^—x-^y-2x-^  by  2a:^2/-^  +  2a:»2^-2-4a?V~'. 

24.  a^«~^  +  2-{-a'^x^  by  2a~^a;^  —  4a"^a;^  -^-^a'^x^. 
26.  3a^6-i  +  a*-2a-*6by  6a^&-i-2a"*-3a-^6. 

225.   The  rule  of  Art.  89  for  the  division  of  exponents 
holds  universally  ;  for,  it  follows  from  Art.  222  that 

—  =  a"*  X  a-**  =  a"*-**.  (Art.  224.) 

-f 
For  example,  —  =  a~^~^  =  cT^ ; 

a~*        -3+1        -JU 

— r  =  a    ^^  =a  ^;  etc. 

EXAMPLES. 
Divide  the  following : 

1.  a«bya-\        4.   a"Hy  a~^.  7.   a-Hy  ~ 

2.  abya^  5.    3c-i  by  ^cr'.  8.    15aby3a-i^6. 

3.  aHya*.         6.   m^  by -^m-^.         9.    Gaj-^^/^  by  3a;2/"^. 


186  ALGEBRA. 


10.   Divide  2a^  —  20  +  ISa"^  by  a  +  2a^-SaK 


2a^  +  4a*-6 


a  +  2a'-3a* 


2a  *  — 4a  ^  —  6a-^,  Ans. 


4a*-14  +  18a"^ 


-4a* 

-    8  +  12a~^ 

-6-12a~*  +  18a"^ 
-6-12a-*  +  18a"^ 

Note.  It  is  important  to  arrange  the  dividend  and  divisor  in  the  same 
order  of  powers,  and  to  keep  this  order  throughout  the  work. 

Divide  the  following : 

11.  a-bhy  a^-b^.  12.   a'^  +  l  by  a"^  + 1. 

13.  x-^-6x-^-4:6-4:0xhy  x'^  +  4. 

14.  x-^—lhyx~^  —  x~^-i-x~^  —  l. 

15.  m  —  Bm^n^-j-Sm^n^—nhym^—n^, 

16.  x-^y~^  -  3  x-^y-^  +  x-^y-^  by  x'^y-^  +  x'^y'"^  -  x-*y-\ 

17.  a  +  ah^  +  bhy  a^  —  a^b^-j-b^. 

18.  m~^  +  m~^n~^  +  n~*  by  m~^  +  m~^n~'^  +  rrT^n'^. 

226.  We  will  now  prove  that  Rule  II.,  Art.  216,  holds 
for  all  values  of  m  and  ti. 

We  will  consider  three  cases,  in  each  of  which  m  may 
have  any  value,  positive  or  negative,  integral  or  fractional. 

Case  I.    Let  n  be  a  positive  integer. 
Then,  from  the  definition  of  a  positive  integral  exponent, 
(a'^Y  =  a"* X  a"* X  a*"  X  •••  to  n  factors 

__  ^TO -j- n* + m +•••  to  w  terms  __  ^wn 


THEORY  OF  EXPONENTS.  187 

Case  II.    Let  n  be  a  positive  fraction,  which  we  will  de- 
note by  -?. 

Then,  (a'")?"=  VJary,  by  the  definition  of  Art.  218, 

=  Vo^,  by  Case  I., 

=  aT,  by  Art.  218, 

=  a«^f . 

Case  III.     Let  n  be  a  negative  quantity,  which  we  will 
denote  by  —  «. 

Then,  (a'»)-  =  ^!— ,  by  the  definition  of  Art.  221, 

=  — ,  by  Cases  I.  or  II., 

We  have  therefore  for  all  values  of  m  and  n, 

(a"*)'»  =  a*"". 
For  example, 

(a-s)"^  =  a~'''~*  =a;  etc. 

EXAMPLES. 
227.   Find  the  values  of  the  following : 

1.  {o?YK       5.  {x-^y\      9.   (-^m3)l        13.  ("— ¥• 

2.  (a-2)2.       e.  (a-^)i      10.   (^r')~'.      14.  (—\^. 

3.  (a3)r         7.  (^i)l.        11.  (^i.^t.  15^  ^[(x-^)^]. 

4.  (c-^)"^.    8.  (Va^)"*.   12.  (xiyl         16.  (a^-^);^. 


188  ALGEBRA. 

228.    To  prove  that  {aby  =  a^b"",  for  any  value  of  n. 

In  Art.   193  we  showed  the  truth  of  the  theorem  for  a 
positive  integral  value  of  n. 

Case  I.    Let  n  be  a  positive  fraction,  which  we  will  denote 
by  5. 

By  Art.  226,  [(a2>)*]*  =  («&)". 

By  Art.  193,  [a*  6^]*    =a^6^  =  (aZ))^ 

Therefore,  [(a6)?"]«=  [afft^J^. 

Taking  the  gth  root  of  both  members, 

p       p  p 
{ab)  9  =aib9. 

Case  II.    Let  n  be  a  negative  quantity,  which  we  will 
denote  by  —  s. 

Then,  (ah) -  =  — ^  =  -L,    by  Art.  193,  or  Case  I. 

'  ^     ^         (aby     a'b'       ^ 

zzza-'b-'. 


MISCELLANEOUS    EXAMPLES.      ^ 
229.    Square  the  following  (Art.  95)  : 
1.    a^-6i  2.   a-^  +  2a.  3.   x'^f-Sx'y-K 

Extract  the  square  roots  of  the  following : 

4.   a-^xK  5.   9mni  6.   ^-^.         7.   ^'. 

^^r  cd*e^ 

10.   a^b~^-4:ah~^-\-6-4:a~h^  +  a-^b^. 


THEORY  OF  EXPONENTS.  189 

Extract  the  cube  roots  of  the  following : 

11.    ah\       12.   -%x-'y^.       13.  %lmH~^,       14.   -^. 

64  a;^ 

15.  82/-2-122/"^'  +  62/"^-2/"^. 

16.  a*  -  9a;^  +  33a;^  -  63a;  +  66a;^  -  36a;^  +  8a;i 

Reduce  the  following  to  their  simplest  forms  : 

1 
17.    a*-«'+2'a2-+i'-3*a'.  20.    [a;-'-"' «^'— *]«-*. 

^  +  2m-r  *  '      \    «"    /  V^'-V 

19.    (aj-)-^-j-(a;-'')-*.  22.    [(aja-tf   -Ja+k. 

23    ^i.^  —  ^^)  -  ^^  (^"^  -  a?"^) . 
2(a*  +  a;*)(a^-a;^) 

24.  ^-^        -  25.    (l-^^^-)'+(^^+^^)' 

26.    (4a;3_3^)(a:2^1)-i^3a;(a^  +  l)f. 

27  (l-3a;  +  a?')--a;(a;-3)(l-3a;  +  a^)"^ 

l-3a;  +  a:2 

28  ^ra?"^+(m  +  a;)"^1   ^       m  +  2a; 

2[a;^  +  (m  +  x)^]         2a;^  (m  +  a;)* 

29.   ^+ri+(l+a^)^P 


190  ALGEBRA. 


XX.    RADICALS. 

230.  A  Radical  is  a  root  of  a  quantity  indicated  by  a  rad- 
ical sign  ;  as,  Va,  or  -vx  +  1. 

If  the  indicated  root  can  be  exactly  obtained,  it  is  called  a 
rational  quantity ;  if  it  cannot  be  exactly  obtained,  it  is 
called  an  irrational  or  surd  quantity. 

231.  The  degree  of  a  radical  is  denoted  by  the  index  of 
the  radical  sign  ;  thus,  Vx-f- 1  is  of  the  third  degree. 

232.  Similar  Radicals  are  those  of  the  same  degree,  and 

5 


with  the  same  quantity  under  the  radical  sign ;   as,  2-\/ax 
and  SVo^. 

233.    Most  problems  in  radicals  depend  for  their  solution 

on  the  following  important  principle  (Art.  228) ; 

11        1 
For  any  value  of  n,    {ah)  «  =  a«  x  h-»-. 

That  is,  ^ab  =  Va  x  Vh. 


TO  REDUCE  A  RADICAL  TO  ITS  SIMPLEST  FORM. 

234.  A  radical  is  in  its  simplest  form  when  the  quantity 
under  the  radical  sign  is  not  a  perfect  power  of  the  degree 
denoted  by  any  factor  of  the  index  of  the  radical,  and  has 
no  factor  which  is  a  perfect  power  of  the  same  degree  as  the 
radical. 

Case  I. 

235.  When  the  quantity  under  the  radical  sign  is  a  perfect 
power  of  the  degree  denoted  by  a  factor  of  the  index, 

1.   Reduce  VS  to  its  simplest  form. 

-^8  =  a/23  =  2^  =  2^  =  V2,*  Ans. 


RADICALS.  191 

EXAMPLES. 
Reduce  the  following  to  their  simplest  forms  : 

2.  \/25.  6.  ^27.  8.  ^64.  11.  ■v/49mV. 

3.  ^9.  6.  ^/lOO.  9.   \/'25^.  12.  -^125 a'b\ 

4.  ^8.  7.  a/81.  10.  ^v/32^.  13.  "Vo^. 

Case  II. 

236.   WTien  the  quantity  under  the  radical  sign  has  a  factor 
which  is  a  perfect  power  of  the  same  degree  as  the  radical. 

1.  Reduce  V54  to  its  simplest  form. 

a/54  =  a/27  X  2  =  a/27  X  a/2  (Art.  233)  =  3  ^2,  Ans, 


2.  Reduce  A/l8a26^  — 27a^6*  to  its  simplest  form. 


Vl8a26^-27a364  =  Vo a25^(2 6  -  3 a) 


A/9a26*X  A/26-3a 


=  3a6^A/26  — 3  a,  ^7i8. 
RULE. 

Resolve  the  quantity  under  t1ie  radical  sign  into  two  factors^ 
one  of  which  is  the  highest  perfect  power  of  the  same  degree  as 
the  radical.  Extract  the  required  root  of  this  factor,  and  pre- 
fix  the  result  to  the  indicated  root  of  the  other. 

Note.  If  the  highest  perfect  power  in  the  numerical  portion  of  thp 
quantity  cannot  be  determined  by  inspection,  it  may  be  found  by  re- 
solving the  number  into  its  prime  factors. 

Thus,  >/1944  =  V¥~xJ^  =  y/Wx^ X  \^"3<3 

=  2  X  32  X  V6  =r  18  V6. 


192 

ALGEBRA. 
EXAMPLES. 

Reduce  the 

following 

to  their  simp 

3.  V50. 

6. 

-5/320. 

4.  3V24. 

5.  V72. 

7. 
8. 

2  a/so. 

■VdSa^l)'. 

12. 

V25a^2/^- 

-50a;y. 

13. 

■i/54ta'b'-\-lS5a^b\ 

9.  a/8T^. 


10.  7V63a^6V. 

11.  -v/250^/?. 


14.    V(a^-2/')(aj  +  2/): 


16.    -Vax^  —  6 ao;  -f  9  a. 


16.    V20ic2+60a;  +  45. 


17.    V3  m^  -  54  m^n  +  243  mnK 

If  the  quantity  under  the  radical  sign  is  a  fraction,  multiply 
both  terms  by  such  a  quantity  as  will  make  the  denominator  a 
perfect  power  of  the  same  degree  as  the  radical.  Then  pro- 
ceed as  before. 

18.   Reduce  \/ — -to  its  simplest  form. 

.|X  =  ^|Ii^  =  J_l_x2a  =  -?-V2^,  Ans, 
\8a^       \16a^       \16a^  4ta^ 

Reduce  the  following  to  their  simplest  forms : 
19. 


20. 
21. 
28. 


^||■     -    «-Alf         '^U 


a^c  —  2  a^bc  -\-  ab^c 


RADICALS.  193 

237.  Conversely^  the  coefficient  of  a  radical  may  be  intro- 
duced under  the  radical  sign  by  raising  it  to  the  power 
denoted  by  the  index. 

I .  Introduce  the  coefficient  of  2  aV^o^  under  the  radical 
sign.  \         

2aV^=  A/8a^x"3^=  \/24aV,  Ans. 

Note.  A  rational  quantity  may  be  expressed  in  the  form  of  a  radical 
by  raising  it  to  the  power  denoted  by  the  index,  and  writing  the  result 
under  the  corresponding  radical  sign. 

EXAMPLES. 

Introduce  the  coefficients  of  the  following  under  the  radical 
signs : 

2.  3V5.        4.  3a/2.        6.  4V5a6.  8.  baV2^. 

3.  2^/7.        5.  4a/5.        7.  a'hV'^K         9.   3mnH*l^'- 

\  27 

10.    (a,_i)J^±I.  12.    Hl^./IE«. 

II.  {l^x)S^^^^^\,        13.   ^^~\l ^- --1. 


ADDITION  AND  SUBTRACTION  OF  RADICALS. 

238.  The  sum  or  difference  of  two  similar  radicals  (Art. 
232)  may  be  found  by  prefixing  the  sum  or  difference  of 
their  coefficients  to  their  common  radical  part. 

1.    Find  the  sum  of  V20  and  V45. 
By  Art  236,        V20  =  Vi^TB  =  2  V5, 
V45  =  V9l<^  =  3V5. 

Hence,     V20  + V45  =  2V5  +  3V5  =  5V5,  Ans. 


194  ALGEBRA. 

2.   Simplify  ^i+^|-^|. 


O  4: 


RULE. 


Reduce  each  radical  to  its  simplest  form.  Unite  the  similar 
radicals,  and  indicate  the  addition  or  subtraction  of  the 
dissimilar, 

EXAMPLES. 
Simplify  the  following : 

3.  y27  +  V12.  9.    V4^+V96^. 

4.  V96  +  V54.  10.    V75  +  V48  -  V245. 

5.  V180-V45.  11.    ^16 +  -^54+ ^128. 

6.  ^162-^48.  l2.-^|_^i+^^. 

7.  V128  +  V98  +  V50.         13.   ^/5  -  ^1  +  ^A. 

15.  7V27-V75-24^i-27^±. 

16.  V27aP  +  V75a3+(a-36)V3a. 


17.    V9a^  +  18«^&- V4:a6«  +  8&'' 


RADICALS.  195 

18.    -^24 +  5^54 --^250 -^192. 


19.    V2Sa^x-28ax-\-7x-^7a^x-{-^2ax  +  6^x. 
20. 


x  —  y,        Ix-^y      Sv'  —  a^    1-3, -^ 


TO  REDUCE  RADICALS  OF  DIFFERENT  DEGREES  TO 
EQUIVALENT  RADICALS  OF  THE  SAME  DEGREE. 

239.  1.  Reduce  V^?  \/3,  and  ^5  to  equivalent  radicals 
of  the  same  degree. 

By  Art  218,  V2  =  2^  =  2^  =  ^2«  =  ^64. 
^3  =  3^  =  Z^  =  ^3*  =  ^81. 
^5  =  5*  =  5T^  =  ^53  =  2J/125. 

RULE. 

Write  the  radicals  with  frax^tional  exponents^  and  reduce 
these  exponents  to  a  common  denominator. 

Note.  The  relative  magnitude  of  radicals  may  be  compared  by  re- 
ducing them  to  the  same  degree.  Thus,  in  Ex.  1,  1^125  is  greater  than 
^^81,  and  ?^81  than  2^64.  Hence,  ^5  is  greater  than  ^3,  and  ^3 
than  v^. 

EXAMPLES. 

Reduce  the  following  to  equivalent  radicals  of  the  same 
degree : 

2.  -^2  and  V^.  5.    a^,  v'36,  and  VTc. 

3.  V^'  -v^^'  ^^^  \/^-  ^-    ^^2/j  '^y^i  and  -Vzx. 

4.  -^5,  ^6,  and  -^7.  7.    Va^h  and  v'o^. 


196  ALGEBRA. 

8.  Which  is  the  greater,  ^2  or  ^3? 

9.  Which  is  the  greater,  -^3  or  ^5  ? 

10.   Arrange  in  order  of  magnitude  -y/3,  ^4,  and  -^7. 

MULTIPLICATION    OF    RADICALS. 
240.   1.  Multiply  V6  by  Vl5. 
By  Art.  233, 

V6  X  Vl5  =  V6"xT5  =  V90  =  3  VlO,  Ans. 

2.  Multiply  V2a  by  VSaK 

Reducing  to  equivalent  radicals  of  the  same  degree, 
V2a  =(2a)*  =(2a)*  =  a/ (2^^  = -^/g^ 

-?/3^  =  (3a2)  3  =  (3a2)^  = -v/(3^  =  ^9^ 

Hence,  V2a  X  -V3a^=  </S^^  X  -\/9^^  = -^/W^' 
=  aV72a,  Ans. 

RULE. 

Reduce  the  radicals  to  equivalent  radicals  of  the  same  degree. 
Multiply  together  the  quantities  under  the  radical  signs,  and 
write  the  product  under  the  common  radical  sign. 

Note.  The  result  should  be  reduced  to  its  simplest  form. 

EXAMPLES. 
Multiply  the  following : 

3.  V6  and  V42.  7.  ?^12  and  ?^2. 

4.  5V10and3V15.  8.  ^2  and  </3. 

5.  2  V3^  and  5  Vl5a;.  9.  Va^  and  Vfe^. 

6.  Va^  and  VaS^.  10.  VTa^  and  V2a. 


RADICALS.  197 

11.  4^3  and  3  V2.  13.    V^,  \/2,  and  J-- 

12.  V^,  V2/2;,  and  -Vzx.  14.    ■\/2x,  -\/'dx,  and  -v/ — s* 

\3ar 

16.    Multiply  2  V3  +  3  V2  by  3  V3  -  V^- 

2V3+3V2 
3V3-    V2 

18  +  9  V6 
-  2  V6  -  6 


18  + 7V6- 6  =  12  +  7  V6,  Am. 

Note.  It  should  be  remembered  that  to  multiply  a  radical  of  the 
second  degree  by  itself  simply  removes  the  radical  sign;  thu«, 
V3X\/3  =  3. 

16.   Multiply  3  Va^+1  +  4a;  by  2  V»^+l  -  x. 

3Vx2+l+4a; 
2  Var^  +  1  —    a; 


6(aj2  +  l)+8a;Var^  +  l 

-3a;\^M^-4a;2 


==20^  +  6  +  6x^^^1,  Arts, 
Multiply  the  following : 

17.  V^  — 2andV^  +  3. 

18.  V5-3V2and2V5+V2. 

19.  V^  -  4  V3  and  2^x  +  ^S. 

20.  2-yya-3-y/b  and  4:  ^a  +  ^b. 

21.  V^— V2/+V^  ^"^  V^+Vy— V*- 


198  ALGEBRA. 

22.  Va;  H-  1  -  2  Vcc  and  2  Vo;  +  1  +  Vic. 

23.  V2- V3  + V5  and  V2  +  V3  +  V^• 
24.    3V5-2V6  +  V7and6V5+4V6-2V7. 

25.    8V3  +  10V2-3V^  and4V3-oV2- V^' 
Expand  the  following  (Art.  95)  : 


26.  (2V3-3)2.  28.   (Vl-a^  +  a)^ 

27.  (3V8  +  5V3)'.  29.   (Va  +  6- Va-6)= 


30.    (Va^  +  l+a;)(Vaj2  +  l-a;), 


31.  ( V^Tl  +  Va;  -  1)  ( Va;  +  1  -  Va;  -  1) . 

32.  (3V2a;  +  5  +  2V3ic-l)(3V2a;  +  5-2V3»-l). 

DIVISION  OF  RADICALS. 
241.   By  Art.  233,     V^^  Va  X  V6. 

Whence,  — —  —  V6. 

RULE. 

Reduce  the  radicals  to  equivalent  radicals  of  the  same  de- 
gree. Divide  the  quantities  under  the  radical  sign,  and  write 
the  quotient  under  the  common  radical  sign. 

EXAMPLES. 
1.   Divide  v^l5  by  V5. 

Reducing  to  equivalent  radicals  of  the  same  degree,  we 
have 

^15      ■v/225 


^125      \l25      \5' 


V^       Vl25 


RADICALS.  199 

Divide  the  following : 

2.  V108  by  V6-  ^           7.  ^2  by  ^3. 

3.  V50?  by  V2c.  8.  ^12  by  ^2. 

4.  -Wa^  by  VSa,  9.  -^/la  by  ^/2~a. 


5.  V6  by  ^3.  10.    ■VSd'b  by  VSa^^d^^ 

6.  ^18  by  V6.      •  11.    a/12^W  by  </2^y?. 

INVOLUTION  AND  EVOLUTION  OF  RADICALS. 

242.  Any  power  or  root  of  a  radical  may  be  found  by 
using  fractional  exponents. 

1.  Raise  ^12  to  the  third  power. 

(^12)3  =  (12^)3  =  12^  =  122  =  ^12  =  2 V3,  Am. 

2.  Raise  -^2  to  the  fourth  power. 

(^2)^  =(2^)*  =2^  =  -^2^  =  -^16  =  2-^2,  Arts. 

Note  1.  The  following  rule  for  the  involution  of  radicals  is  evi- 
dent from  the  above : 

If  possible,  divide  the  index  by  the  exponent  of  the  required  power-, 
otherwise,  raise  the  quantity  under  the  radical  sign  to  the  required  power. 

EXAMPLES. 
Find  the  values  of  the  following : 
3.    {-s/b)K  6.    (^18)^  9.    (^/32)». 


4.  (V7)2.  7.    {-Va-hy.  10.    (3aV6a;)*. 

5.  (■v^)^  8.    (4V3^)^  11.    (3-v^^4^)2. 


200  ALGEBRA. 

12.  Extract  the  cube  root  of  -\/21^. 

-v/(  V27V)  =  (  V27^)*  =  (  V(3^«)^  =  [(3a;)  ^]^ 
=  (3a;)^  =  V3^,  Ans. 

13.  Extract  the  square  root  of  -^6. 

V(-^6)  =  (6^)^  =  6*  =  ^6,  Ans. 

Note  2.  The  following  rule  for  the  evolution  of  radicals  is  evident 
from  the  above : 

If  possible,  extract  the  required  root  of  the  quantity  under  the  radical 
sign;  otherwise,  multiply  the  index  of  the  radical  by  the  index  of  the 
required  root. 

K  the  radical  has  a  coefficient  which  is  not  a  perfect  power  of  the 
same  degree  as  the  required  root,  it  should  be  introduced  under  the 
radical  sign  before  applying  the  rule.     Thus, 

Find  the  values  of  the  following ; 

14.  V(V2).  17.    V{V21^').  20.  ^(3V3). 

15.  ■^(Vl25).       18.    •^(■</^qr5).  21.  -v^C^^). 

16.  ^/(V32).        19.    ^{■Vx'-2x-^l).  22.  V{W2). 


TO    REDUCE    A    FRACTION    HAVING    AN    IRRATIONAL 

DENOMINATOR  TO  AN  EQUIVALENT  FRACTION 

WHOSE  DENOMINATOR  IS  RATIONAL. 

Case  I. 

243.    When  the  denominator  is  a  monomial. 

The  reduction  is  effected  by  multiplying  both  terms  by  a 
radical  of  the  same  degree  as  the  denominator,  having  such 
a  quantity  under  the  radical  sign  as  will  make  the  denomi- 
nator of  the  resulting  fraction  rational. 


RADICALS.  201 


5 
1.   Reduce  — ; —  to  an  equivalent  fraction  with  a  rational 


,  Ans. 


denominator. 

Multiplying  both  terms  by  V9  a, 

■V3a^~  -VSaF ■y/^~  ^/27^^~    3a 

EXAMPLES. 

Reduce  the  following  to  equivalent  fractions  with  rational 
denominators : 

2.   A.  4.   -l£-.  6.        1 


V2  V5^  </T6^ 

3.  -i-.  5.   -4..  7.   -4^. 

\/4  -y/da'  ^/27a' 


Case  II. 

244.    Wien  the  denominator  is  a  binomial^  containing  rad- 
icals of  the  second  degree  only. 

-y/5    -v/2 

1.   Reduce  to   an   equivalent  fraction   with    a 

rational  denominator. 

Multiplying  both  terms  by  Vo  —  V2, 

V5-V2^  (V5-V2)2  ^5-2VlO  +  2 

V5+V2      (V5+V^2)(V5-V2)  5-2 


2.   Reduce  —^ — =  to  an  equivalent  fraction  with  a 
l_Vl-a; 
rational  denominator. 


202  ALGEBRA. 


Multiplying  both  terms  by  1  +  Vl  —  ic, 


1-Vl-a;      (l_Vl-aJ)(l  +  Vl-a;) 


l-(l-a.) 


2-.+  2V1-.    ^^^^ 


RULE. 


Multiply  both  terms  of  the  fraction  by  the  denominator  with 
the  sign  between  its  terms  changed. 


EXAMPLES. 

Reduce  the  following  to  equivalent  fractions  with  rational 
denominators : 


3. 


4  y     2V5+V2  jj     g-Va^-l 

3+V2  *    V5-3\/2  '    a  +  Va^-l 


2-V3  *    a+Vic  ic-Va^-4 


5     V2— V3        g     Va  +  1  —2         jn     Va+a;+V«— a? 
V2  +  V3  Va-fl  —  1  Va+ac"—  Va—x 

g     Va+V&.      jQ     Va;  +  2— Va;      ^^     Va^— 1  — Va^+1. 
Va—-Vb  '    Vx+^+Vx  Va^-l  +  Va^+l 

245.  The  approximate  value  of  a  fraction,  whose  denomi- 
nator is  irrational,  may  be  most  conveniently  found  by 
reducing  it  to  an  equivalent  fraction  with  a  rational  denomi- 
nator. 


RADICALS.  203 


1.   Find  the  approximate  value  oi to  three  places 

of  decimals.  ""  ^ 


2+V2 


2-V2      (2-v2)(2+V2) 
^2+V2 
4-2 

=  ^  +  ^'^^^•'•  =  1.707...,^... 
2  ' 


EXAMPLES. 

Find  the  approximate  values  of   the  following  to  three 
decimal  places  : 

2        3  3   _!_.        ±,  V3-V2         g    2V5-V3 


V2-1  ^9  V3  +  V2  3V5+2v3 

IMAGINARY  QUANTITIES. 

246.  An  Imaginary  Quantity  is  an  indicated  even  root  of 
a  negative  quantity  (Art.  201)  ;  as,  V— 4,  or  V  — a^. 

In  contradistinction,  all  other  quantities,  rational  or  irra- 
tional, are  called  real  quantities. 

247.  Every  imaginary  square  root  can  be  expressed  as 
the  product  of  a  real  quantity  multiplied  b}'  V—  1.     Thus, 


V-a2  =  Va2x(-l)  =  Va2xV-l  =  aV-l; 


V-5  =V5   x(-l)  =  V5V-l;  etc. 

248.  Let  it  be  required  to  find  the  powers  of  V—  1 . 

By   Art.    198,    V—  1    signifies   a   quantity   which,  when 
multiplied  by  itself,  will  produce  —  1  ;  that  is, 


204 


ALGEBRA. 


Therefore, 

(V^)'=(V^)^X   V^   =      1  xV'=a  =  V^;etc. 

Thus  the  first  four  powers  of  V  — 1  are  V  — 1,  —  1,  —  V  — 1, 
and  1 ;  and  for  higher  powers  these  terms  recur  in  the  same 
order. 


MULTIPLICATION  Or  IMAGINARY  QUANTITIES. 

249.   The  product  of  two  or  more  imaginary  square  roots 
may  be  found  by  aid  of  the  principles  of  Arts.  247  and  248. 

1.  Multiply  V^  by  V^. 
By  Art.  247, 

V^xV^=V2V^i^xV3V^ 

=  V2V3(-\/^)2 

=  V6x(-l)  (Art.  248)  =  -V6,  Aug. 


2.  Multiply  V"=^S  V-6S  and  V-c^ 

V^'xV^2xV^r^  =  aV^x6V^xcV^ 

=  a6c(V  — 1)^  =  — a&cV  — 1,  Ans. 


RULE. 


Reduce  each  imaginary  quantity  to  the  form  of  a  real  quan- 
tity multiplied  hy  V—  1.  Form  the  product  of  the  real  quan- 
tities^ and  multiply  the  result  hy  the  required  power  of  V—  1. 


RADICALS. 


205 


EXAMPLES. 
Multiply  the  following : 

3.  4  V^  and  2  V^^.  6.  V^,  V^,  and  V^^. 

4.  V^^  and  V^=^.  7.  1-2  V^  and  3  +  V^. 
6.    -3V'^and4V^.         8.  4  + V^  and  8-2  V^. 

9.    2V^^-3V'=^and4V^^  +  6V^. 
10.    V^,  V^,  V^^^,  and  V-25. 

Expand  the  following : 

11.  (2_V33)2.  13.   (1  +  V^)(1-V^). 

12.  (V^  +  2V^^2y.  14.   (a  +  V^)(a-V^=^). 

16.   (a;V^^  +  2/V^)(x'V^^  — 2/V'— 2/). 
16.   (1  +  V^)2  +  (1-V^)^ 
17.  Divide  V— a;by  V^^. 

V— a;      Va;  V  — 1      Va^ 


V— 2/     Vt/V^     V.y 


Vf 


Ans. 


Note.  The  rule  of  Art.  241  would  have  given  the  same  result; 
hence,  that  rule  applies  to  the  division  of  all  radicals,  whether  real  or 
imaginary. 


Divide  the  following : 

18.  V"^  by  V^^. 

19.  V^=^  bv  V^=^. 


20.    ^y=T2  by  V^. 


21.    V-^by  V-2. 


206  ALGEBRA. 

PROPERTIES  OF  QUADRATIC  SURDS. 

250.  A  Quadratic  Surd  is  the  indicated  square  root  of  an 
imperfect  square  ;  as,  -y/S,  or  -^7, 

251.  A  quadratic  surd  cannot  be  equal  to  a  rational  quan- 
tity plus  a  quadratic  surd. 

For,  if  possible,  let        -^/a  =b+  ^c. 

Squaring  the  equation,      a  =  6^  +  2  6  y'c  +  c. 

Or,  2b-y/c  =  a-b^-c,  . 

Whence,  -^c  =     ~     ~   - 

ij  0 

That  is,  a  surd  equal  to  a  rational  quantity,  which  is 
impossible.     Hence  -^Ja  cannot  be  equal  to  &  +  -yjc, 

262.    To  prove  that  if  a-\-  -yjb  =  c-\--yJd^  then  a  =  c,  and 
^br=^d. 

If  a  is  not  equal  to  c,  let  a  =  c  +  x.     Substituting,  we  have 

c  +  x-\-^b  =  c  +  -^d. 

Or,  a;  4-^6  =  yd, 

which  is  impossible  by  Art.  251.     Hence  a  =  c,  and  conse- 
quently -y/b  =  ^d. 

253.    To  prove  that  if  Va  +  Vb  =  -Vx  +  Vy,  then  Va  —  ^/b 

=  -yyx-y/y. 

Squaring  the  equation  ■\/a4-  y/b  =  Vx  -}-  Vy, 

we  have  a  +  Vb  =  x  +  2  Vxy  +  y. 

Whence,  by  Art.  252,  a  =  x-\-y,               (1) 

and  Vb  =  2Vxy.              (2) 

Subti'acting  (2)  from  (1) ,  a  —  Vft  =  x  —  2  Vxy  +  y. 


Extracting  the  square  root,  Va—  y/b  =  Va;  —  Vy. 


RADICALS.  207 


SQUARE  ROOT  OF  A  BINOMIAL  SURD. 

254.   The  preceding  principles  serve  to  extract  the  square 
root  of  a  binomial  surd  whose  first  term  is  rational. 

For  example,  required  the  square  root  of  13  —  ^160. 


Assume  Vl3- V160=  Va;- Vy.  (1) 


Then,  by  Art.  253,        V13  +  V160  =  Vic  +  Vy.  (2) 


Multiplying  (1)  by  (2),  a/169-160  =  a; -  ?/. 
Or,  a; -2/ =  3.  (3) 

Squaring  (1),  13-VlQ0  =  x—2^/xy■j-y. 

Whence,  by  Art.  252,  a;  +  2/  =  13.  (4) 

Adding  (3)  and  (4),  2a;  =16,  or  a;  =  8. 

Subtracting  (3)  from  (4),  2y  =  10,  or  y  =  5. 

Substituting  in  (1),        ■\/l3-V160  =  VS  -  V5 

=  2V2-V5,  Ans, 

255.  Examples  like  the  above  may  often  be  solved  by 
inspection  by  expressing  the  given  quantity  in  the  form  of  a 
perfect  trinomial  square  (Art.  108) ,  as  follows  : 

Reduce  the  surd  term  so  that  its  coefficient  may  be  2.  Sepa- 
rate tJie  rational  term  into  two  parts  wJiose  product  is  the 
quantity  under  the  radical  sign.  Extract  the  square  roots  of 
these  pa7'ts,  and  connect  them  by  the  sign  of  the  surd  term. 

1.   Extract  the  square  root  of  8  +  V48. 


V8+V48=:V8  +  2V12. 

We  then  separate  8  into  two  parts  whose  product  is  12. 
The  parts  are  6  and  2  ;  hence, 


Vs  -i- 2 V 1 2  =  V6  +  2V'6>^  +  2 
:=V6  +  V2,  Ans. 


208  ALGEBRA. 

2.   Extract  the  square  root  of  22  —  3  V32. 

V22  -  3  ^732  =  V22  -  v'288  =  V22  -  2  V72. 

We  then  separate  22  into  two  parts  whose  product  is  72c 
The  parts  are  18  and  4  ;  hence, 


V22  -  3  V32  =  Vl8-2V72  +  4 

=  Vl8-V4  =  3V2~2,  Ans, 

EXAMPLES. 
256.   Extract  the  square  roots  of  the  following : 

1.  12  +  2V35.  6.  8-V60.  11.  23+V360. 

2.  7-2V12.  7.  15  +  4V14.  12.  24-2V63. 

3.  9  +  2VS-  8.  12~-V108.  13.  33+20V2. 

4.  9-4V5.              9.  20-5V12.  14.  47-6V10. 
6.  16  +  6V7.           10.  14  +  3V20.  15.  67-7V72. 

16.  2m~2Vm2-7i2.  17.  2 a  +  x -{- 2  VoTfax* 

SOLUTION  OF  EQUATIONS  CONTAINING  RADICALS. 


257.  1.  Solve  the  equation  Var^  — 5  —  a;  =  —  1 , 


Transposing,  V  flr*  —  5  =  a;  —  1 . 

Squaring  both  members,         a^  —  5  =  a^— 2a;-f-l. 
Transposing  and  uniting  terms,  2a;  =  6. 

x  —  3,  Ans. 


RADICALS.  209 


2.  Solve  the  equation  V2a;— 1  -f  V2a; -f  6  =  7. 


Transposing  y/2x  —  1 , 


V2a;-f6  =  7-V2a;-l. 


Squaring,  2a;  +  6  =  49  - 14  V2a;-1  +  2a; -  1. 

Transposing  and  uniting, 


14V2a;-l  =  42. 
Or,  V2^^  =  3. 

Squaring,  2  a;  — 1  =  9. 

2a;=10. 
a;  =5,  -4715. 

RULE. 

Transpose  the  terms  of  the  equation  so  that  a  radical  term 
may  stand  alone  in  one  member;  then  raise  both  members  to 
a  power  of  the  same  degree  as  the  radical. 

If  there  are  still  radical  terms  remaining ,  repeat  the  opera- 
tion. 

Note.  The  equation  should  be  simplified  as  much  as  possible  before 
performing  the  involution. 

EXAMPLES. 

3.  V5^^-2  =  l.  8.  i/a^-6x'-x-{-2  =  0. 

4.  6=-\/2x-\-S.  9.  Va;  +  Va;  +  5  =  5. 

6.  A/4a; 4-3  =  3.  10.  Va;- 32  +  Va;=  16. 


6.  V4a;2_i9_2a;=  -1.  11.   Va;-3  -  Va;-t-12  =  -3. 


7.  Va;2-3a;  +  6  =  2-a;.  12.  V2a;-7+ V2a;  +  9  =  8. 


210  ALGEBRA. 


13.  V3a;  +  10~V3a;  +  25  =  -3. 

14.  ^{x-ay  +  2ab-{-b"-  =  x-a  +  b, 

15.  Va^-3a;  +  5-Vaj2-5a;-2  =  l. 

16.  Vx-V^':r3  =  _2_. 

17.  Vx— l+Vi»  +  4  =  V4a;  +  5. 

18.  V«2  +  4a;  +  12-fVar^- 12a;- 20  =  8. 


19. 


Va;  —  3  _  Va;  —  4 


20.  V3^+V3a;H-13  =  -^ 


91 


V3X+13 

21.  Va;  +  l+Va;-2-V4x-3=0. 

22.  Va;4-V^T^=     ^^     . 

Vic  +  a 

23.  Vl9  +  icV^^^=^J  =  a;-3. 


24. 


a 


a; 


=  V5  — ic. 


Va  — a;      V6  — a; 
25.  Va;  +  a+Va;  +  6  =  V4a;  +  a  +  3&. 


26.  Vll+ajVa^+lBH^^  +  l- 

27.  V{a^-2aa;4-a;^V3a-a;i  =  «-^' 


QUADRATIC  EQUATIONS.  211 


XXL   QUADRATIC  EQUATIONS.' 

258.  A  Q,uadratic  Equation,  or  an  equation  of  the  second 
degree  (Art.  167),  is  one  in  which  the  square  is  the  highest 
power  of  the  unknown  quantity. 

A  Pure  Quadratic  Equation  is  one  which  contains  only  the 
square  of  the  unknown  quantit}^ ;  as,  aaP  =  b. 

An  Affected  Quadratic  Equation  is  one  which  contains 
both  the  square  and  the  fust  power  of  the  unknown  quan- 
tity ;  as,  aic^  -j-  6ic  -H  c  =  0. 

PURE  QUADRATIC  EQUATIONS. 

259.  A  pure  quadratic  equation  is  solved  by  reducing  it 
to  the  form  sc^  =  a,  and  then  extracting  the  square  roots  of 
both  members. 

1.  Solve  the  equation    Sx^-}-7  =— -\-35. 

Clearing  of  fractions,  12ic2  +  28  =  5a^+  140. 

Transposing  and  uniting,      7a^=112. 

Or,  x'^W. 

Taking  the  square  root  of  both  members, 

a;  =  ±4,  Ans. 

Note  1.  The  double  sign  is  placed  before  the  result  because  the 
square  root  of  a  number  is  either  positive  or  negative  (Art.  201). 

2.  Solve  the  equation     Tar'  —  5  =  5 x^  —  13. 
Transposing  and  uniting,       2a^  =  —  8. 

Or,  '  a^  =  -4. 

Whence,  a;=±V— 4 

Note  2.  Since  the  square  root  of  a  negative  quantity  is  imaginary 
(Art.  246),  the  values  of  x  can  only  be  indicated. 


212  ALGEBRA. 

EXAMPLES. 
Solve  the  following  equations  : 


3.  4:3^-7=29.  6.   4-V3a^  +  16^=6. 

4.  bx'  +  b^Sx'-^-bb,  7.   ax^  +  b  =  c. 
g_5 L  =  _M.                 8        ^         ^         ^ 


6ic2     4a^  16  4-x     3      4  +  a; 

9     2(x  +  3)  (aj  -  3)  =  (ic  +  1)^  -  2ic. 

10.  (3  a;  -  2)  (2 a;  +  5)  +  (5 a;  +  1)  (4a;  -  3)  -  91  =  0. 

11.  ^_34.i^  =  JL_a^  +  §M. 

2  12       24  24 

12     2a7^-5      3a;^4-2      a^-lQ^Q 
3  7  6  * 

13        <^      ^      6  14     4a;^-3  ^  2(9a^  +  2) 

ar^-6      a^-a*  '    2a:2_i      3(3a^+2) 

16.    (2 a;  -  a)  (a;  -  6)  +  (2  a;  +  a)  (a;  +  &)  =  a^  +  b\ 

jg     5a^-l      3a^H-l  89  _« 


a^-3        a;2_|_2       (a^_  3)(a;2^  2) 
17.    a;  +  V^T3  = 

18.  L= 1 =:^. 

l_Vl-a;^      1  +  Vl-a^       a^ 

AFFECTED  QUADRATIC  EQUATIONS. 

260.  An  affected  quadratic  equation  is  solved  by  adding 
to  both  members  such  a  quantity  as  will  make  the  first  mem- 
ber a  perfect  square  ;  an  operation  which  is  termed  complet- 
ing the  square. 


QUADRATIC   EQUATIONS.  213 


FIRST  METHOD  OF  COMPLETING  THE  SQUARE. 

261.  Every  affected  quadratic  equation  can  be  reduced  to 
the  form 

^  +px  =  q ; 

where  p  and  q  represent  any  quantities  whatever,  positive  or 
negative,  integral  or  fractional. 

Let  it  be  required  to  solve  the  equation  a^ -\-3x  =  4:. 

In  any  trinomial  square  (Art.  108),  the  middle  term  is 
twice  the  product  of  the  square  roots  of  the  first  and  third 
terms ;  hence  the  square  root  of  the  third  term  is  equal  to 
the  second  term  divided  by  twice  the  square  root  of  the  first. 

Therefore  the  square  root  of  the  quantity  which  must  be 

added  to  or +  3 a;  to  make  it  a  perfect  square,  is  — ,  or  — . 

Zx  A 

3        9 
Adding  to  both  members  the  square  of  - ,  or  -,  we  have 

iB2  +  3aj  +  -  =  4-|--  =  — • 
4  4       4 

Extracting  the  square  root  of  both  members, 

a;  +  -  =  ±  — 
2  2 

Transposing  |,  a;  =  _  |  + 1,  or  - 1  - 1. 

Whence,  x~\  or  —4,  Ans. 

2Q2l.  From  the  above  operation  we  derive  the  following 
rule  : 

Reduce  the  equation  to  the  form  oc^  +px  —  q. 

Complete  the  square  by  adding  to  both  members  the  square 
of  half  the  coefficient  of  x. 

Extract  the  square  root  of  both  members^  ciud  solve  the  sim- 
ple equation  thus  formecl,, 


214  ALGEBRA. 

1.  Solve  the  equation  da^  —  8x=  —  4:. 

Dividing  by  3,  a^-^  =  -i, 

3  3 

which  is  in  the  form    a^  +  pa?  =  g. 

Adding  to  both  members  the  square  of  -,  or  — , 

3  9' 

3  "^  9  3"^  9       9' 

Extracting  the  square  root, 

3  3 

Whence,  a-^^l-t^gor?,  Ans, 

3      3  3 

Note.  These  values  may  be  verified  as  follows : 
Putting  a:  =  2  in  the  given  equation,  12  —  16  =  —  4. 

Puttinga:=?,  4_16^_^ 

3  -^  3      3  * 

If  the  coefficient  of  a^  is  negative,  it  is  necessary  to  change 
the  sign  of  each  term. 

2.  Solve  the  equation  — 3a^— 7ic  =  — • 
Dividing  by  -  3,     x^-\-~  =  -~ 

Adding  to  both  members  the  square  of  -,  or  — , 

6         36 

3       36  9       36      36* 

Extracting  the  square  root, 

6  6 

Whence,  (c=-^  ±  ?  =  -?  or -5,  ^^5. 

6      6  3  3 


QUADRATIC  EQUATIONS.  215 

EXAMPLES. 
Solve  the  following  equations : 

3.  a^4-4a;=5.  8.    2iK2  +  5a;  =  -2. 

4.  ic2-5a;  =  -4.  9.    4:aF-8x  +  S  =  0, 
6.   a^_7a;  =  -12.                      10.    4a^-3  =  llx. 

6.  a^-\-x=6.  11.   S-x-2x^  =  0. 

7.  3a^_4aj=:4.  12.    14  4-15a?-9«2=:0. 

263.  If  the  coefficient  of  a^  is  a  perfect  square,  it  is  con- 
venient to  complete  the  square  directly  by  the  principle  of 
Art.  261  ;  that  is,  by  adding  to  both  members  the  square  oj 
the  quotient  obtained  by  dividing  the  second  term  by  twice  the 
square  root  of  the  first. 

1.    Solve  the  equation  9  a;^  —  5  a;  =  4. 

The  quotient  of   the  second  term  divided  by  twice  the 

5  5 
square  root  of  the  first,  is  -.     Adding  the  square  of  -  to 

both  members, 

9a--5a.+?^=4  +  ?5  =  i5?. 
36  36       36 

Extracting  the  square  root, 

Q        5       .  13 
3a; ==  ±  — 

6  6 

6       6  3 

Whence,  a;  =  l  or  — -,  Ans* 

«/ 

Note.  K  the  coefficient  of  x^  is  not  a  perfect  square,  it  may  be 
made  so  by  multiplication. 

Thus,  in  the  equation  18  x'^  +  5  a:  =  2,  the  coefficient  of  x^  may  be 
made  a  perfect  square  by  multiplying  each  term  by  2. 

If  the  coefficient  of  x"^  is  negative,  the  sign  of  each  term  must  be 
changed. 


216  ALGEBRA. 

EXAMPLES. 
Solve  the  following  equations : 

2.  4a^  +  3ic=10.  7.  8a^  +  a;-34  =  0. 

3.  9ic2+2ic=ll.  8.  11x^12-360^  =  0, 

4.  25a^-15a;  =  -2.  9.  6x'-bx  =  -l. 

5.  4a^-7a;=-3.  10.  32a;2  +  20£C- 7  =  0. 

6.  2a;2  +  15a;  =  -13.  11.  48a^-32aj  =  3. 

SECOND  METHOD  OF  COMPLETING  THE  SQUARE. 
264.   Every  affected  quadratic  can  be  reduced  to  the  form 

aaP  -\-bx  =  c. 
Multiplying  each  term  by  4  a,  we  have 

4  a^ic^  H- 4  a6a;  =  4  ac. 

Completing  the  square  by  adding  to   both   members   the 
square  of  b  (Art.  263), 

4aV  +  4a6ic  +  &2  =  62  4- 4a<j. 
Extracting  the  square  root, 


Transposing, 
Dividing  by  2  a, 


2aa;  +  &  = 

±  V62  +  4ac. 

2aa;  = 

-6±V6'  +  4ac. 

/y?  — 

-6±V6^  +  4ac 

2a 


265.  From  the  above  operation  we  derive  the  following 
rule : 

Reduce  the  equation  to  the  form  ao?  ■\-hx  =  c. 

Multiply  both  members  by  four  times  the  coefficient  of  oi?^ 
and  add  to  each  the  square  of  the  coefficient  of  x  in  the  given 
equation. 

Extract  the  square  root  of  both  members,  and  solve  the  sim- 
ple equation  thus  formed. 


QUADRATIC   EQUATIONS.  217 

Not&  The  advantage  of  this  method  over  the  preceding  is  in  avoid- 
ing fractions  in  completing  the  square. 

1.  Solve  the  equation  2 a^  —  7a;  =  —  3. 
Multiplying  both  members  by  4  times  2,  or  8, 

Adding  to  each  member  the  square  of  7, 

1 6 ic2  _  5g  3.  ^  49  ^  _  24  _^  49  =  25. 

Extracting  the  square  root, 

4a;-7  =  ±5. 
Transposing,  4a;=7±5  =  12crr2. 

Dividing  by  4,  x  =  3  or  -,  A^is. 

If  the  coefficient  of  x  in  the  given  equation  is  an  even 
number,  fractions  may  be  avoided,  and  the  rule  modified,  as 
follows : 

Multiply  both  members  by  the  coefficient  of  ar',  and  add  to 
each  the  square  of  half  the  coefficient  of  x  in  the  given  equa- 
tion. 

2.  Solve  the  equation  7  ar  +  4  a;  =  51. 
Multiplying  both  members  by  7, 

49x'2  +  28a;  =  357. 
Adding  to  each  member  the  square  of  2, 

49x2_^28a;  +  4  =  361. 
Extracting  the  square  root, 

7a;  +  2  =  ±19. 

7a;=-2±19=:17or  -21. 

17 
Whence,  x  =  —  or  —3,  Ans. 


218  ALGEBRA. 

EXAMPLES. 
Solve  the  following  equations  : 

3.  2ic2  +  5a;=3.  10.  17a;  +  20  =  -3a^. 

4.  Ax'-x^S.  11.  5x'-S  =  Ux. 

5.  a^-3a;=18.  12.  2  +  x-6a^  =  0. 

6.  3a^H-4a;  =  4.  13.  Saj^^- 60;  + 1  =0. 

7.  8a^  +  2a;  =  3.  14.  7x  +  3  =  6a^. 

8.  2ic2-7a;=15.  15.  15a^-8a;  =  -l. 

9.  7a^-16a;+'4=0.  16.  41a;- U- 15a^  =  0. 

MISCELLANEOUS  EXAMPLES. 

266.  The  following  equations  may  be  solved  by  either  of 
the  preceding  methods ;  preference  being  given  to  the  one 
best  adapted  to  the  example  considered. 

1.  ^  +  ^  +  J_  =  o.  3.  A  =  _L_?. 

2      3      24  2x      6af      3 

*    a;'^2  2*  '    5      2x         4:0?' 

5.  {x  +  6)(x-5)-{nx-\-l)=0, 

6.  4a;(18a;-l)  =  (10a;-l)2. 

7.  (3x-5y-(x  +  2y  =  -5. 

8.  (a;  4- 3)3 -(ic- 1)^=19. 

9.  (a; -1)2 -(3a; +  8)2 -(2a; +  5)2  =  0. 

^Q     2_x±S_2_x±9^^^  ^2     4^-ii^^=14. 

8  +  a;       3a;  +  4  a;+l 

11     5      3a;  +  l^l  ,«       21        a;  ^25 

*    a;  x"          4*  '    5-a;      7       7* 


14. 

3ic2       l-Sx     X 
x-1         10          5 

15. 

X         a;-  1      3 
x-l          X         2 

16. 

X         b  —  x      15 

b  —  x         X          4 

QUADRATIC  EQUATIONS.  219 


17     ^  +  1  _  a^  +  3  ^  8 

a; +  2      a; +  4  "3' 


18.    V20-ha;-ar^  =  2a;-10. 


19.    2VaJ+— =5. 


20.  ^^^^ 3^+1  ==0. 

a;         3a;-l      2 

21.  ^^^±l=.  +  il. 
a^+3x--l  3 

oo     2a^+3a;-5       2ar'-a;-l 


23. 


3a^_|_4a;_l      3a^-2a;  +  7 

7  3     ^22 

ar^  -  4      a;  +  2       5  * 


24.   -^  +  1  1         ■       1 


a^-1      3      3(a;-l)      a;  +  l 

25     ^  +  3  ,  a;  — 3_2a;  — 3 
a;  +  2      a;— 2       aj— 1 


2g     12  +  5a;  ,  2-\-x_      1 


12  — 5a;         a;  1  — 5a; 


27.  V4a;-3-Va;H-l  =  l. 

28.  2Va;  =  Va;  +  5H — -^ 

Va;  +  5 

29     a;  +  2_2a;+16      a;  — 2 
'    x—l        x-\-b        x-\-l 


30.    V3a;  +  l+V2a;-l=V9a;  +  4, 


220  ALGEBRA. 

The  same  methods  are  applicable  to  the  solution  of  literal 
quadratic  equations. 

31.  Solve  the  equation  a;^  —  2  mx  =  2  m  +  1 . 
Completing  the  square  by  adding  m^  to  both  members, 

x^  —  2mx-\-m^  =  m^  +  2m-\-l={m-\-iy. 
Extracting  the  square  root, 

Whence,  a;  =  mH-(mH-l),  or  ??i— (m  +  1) 

=  2m -h  1  or  —  1,  Ans. 

32.  Solve  the  equation  a^  +  aa;  —  6aj  —  a6  =  0. 
The  equation  may  be  written, 

a^  +  (<^  —  h)x  —  ab. 
Multiplying  both  members  by  4  times  the  coefficient  of  a^, 

4a^4-4(a-6)x  =  4a6. 
Adding  to  each  member  the  square  of  a  —  6, 
4a^  +  4(a  -  h)x-^{a  -  by  =  (a  -  by  +  4a6 

=  a^-i-2ab  +  b\ 
Extracting  the  square  root, 

2x  +  (a-b)  =  ±{a  +  b), 
Whence,  2x  =  -{a-b)±(a  +  b). 

Hence,  2a;  =  — a  +  6  +  a  +  ?)  =  26, 

or,  2x  =  —  a-\-b  —  a  —  b  =  —  2a. 

Dividing  by  2,  a;  =  —  a  or  6,  Ans, 

Note.  If  several  terms  contain  the  same  power  of  x,  the  coefficient 
of  that  power  should  be  placed  in  a  parenthesis,  as  shown  in  Ex.  32. 

Solve  the  following  equations  : 

33.  a^-2aa;=(&+a)(6-a). 

34.  aP  —  ax-\-bx  =  ab. 

35.  x^  —  {a-{-l)x  =  -a. 


QUADRATIC  EQUATIO:S^S.  221 

36.  a^4-2(c  +  8)a;  =  -32c. 

37.  a?  —  m'^{l—m)x  =  m^. 

38.  aca^  —  hex  —  adx  =  —  hd, 

39.  {x  +  ^py  =  {x-{-py  +  Zlp\ 

40.  Q:x?-\-^ax-\~2hx  =  -Za^, 

41.  M2lI1^^^_.  43.   a:+l  =  «  .  ft. 

3a-2a;       4  ic      6      a 

42.  ^    ^    '^V  44.    a;+l^a+l 

'    a;  + 1      wi  + 1  '     V^         V«  * 


45.  V(a  +  6)a;  — 4a6  =  a;  — 26. 

46.  v^z:4^=(«+ft)(«-ft). 

47.  2V?^::^  +  3V2^=^^?^±^. 

48.  ^ 4- ] =1+^. 

a  -\-  Va^  —  X     a  —  Va^  —  x  « 

49.  Va;  +  a+Va;  +  2a  =  V2a;  +  3a. 

50.  a^4-l^a  +  6  I      c 


i8  c         a  +  6 

51.  _4_  =  i  +  i  +  i. 

CI  +  0  4-a;      a      h     x 

52.  a^  4-  Z>a;  +  ca;  =  (a  +  c)  (a  -  6) . 

53.  a6x^  +  ^:=^^^  +  ^^-2^^-^. 

c  c^  c 

54.  (3a2  +  62)(a^_a;  +  i)  =  (362  +  a2)(a^  +  a;4-l). 


222  ALGEBRA. 

SOLUTION  OF  QUADRATIC  EQUATIONS  BY  A  FORMULA. 
267.   It  was  shown  in  Art.  264  that  if  aoi^  +  bx  =  c,  then 


5±VF-+4ac^ 

2a  ^  ^ 

This  result  may  be  used  as  a  formula  for  the  solution  of 
quadratic  equations,  as  follows  : 

1 .    Solve  the  equation  3a:^  +  5x=12. 

In  this   case,   a  =  3,   6  =  5,   c  =  12;     substituting   these 
values  in  (1), 

_  -  5  ±  V25  4- 144  _  -  5  ±  ^169 

X  — —  — — — 

6  6 

-5  ±13  o        4      . 

3  or  -,  Alls. 


6  3 

2.  Solve  the  equation  110cc2-21a;=-l. 

In  this  case,  a=110,  &  =  —  21,  c  =  —  1;  therefore, 

^      21±V441-440      21  ±1       1  1      J 

X  = = =  —  or  — ,  A71S. 

220  220        10         11 

Note.  Particular  attention  must  be  paid  to  the  signs  of  the  coeffi- 
cients in  substituting. 

EXAMPLES. 

Solve  the  following  equations  : 

3.  2iB2  +  5»=18.  8.    5a^-lla;  =  -2. 

4.  Sa^^2x  =  5.  9.   4:a^-Sx-5  =  0. 
6.   ar^-7a;  =  -10.                  10.    6a^  +  25a;  +  14  =  0. 

6.  bx'  +  x^lS,  11.    30x-16  =  dx', 

7.  6ic2  +  7a;  =  -l.  12.    27  +  39a;- lOa;^^  0. 


QUADRATIC  EQUATIONS.  223 

XXII.  PROBLEMS. 

INVOLVING  QUADRATIC  EQUATIONS. 

268.  1.  A  man  sold  a  watch  for  $21,  and  lost  as  much 
per  cent  as  the  watch  cost  him.  Required  the  cost  of  the 
watch. 

Let  X  =  the  cost  in  dollars. 

Then,  x  =  the  loss  per  cent, 

and  X  X  —  =  -^  =  the  loss  in  dollars. 

100     100 

By  the  conditions,       -— -  =  a:  —  21. 
100 

Solving  this  equation,     x  =  70  or  30. 

That  is,  the  cost  of  the  watch  was  either  $  70  or  1 30 ;  for  each 
of  these  values  satisfies  the  given  conditions. 

2.  A  farmer  bought  some  sheep  for  $72.  If  he  had 
bought  6  more  for  the  same  money,  they  would  have  cost 
$  1  apiece  less.     How  many  did  he  buy  ? 

Let  X  =  the  number  bought. 

72 
Then,  — =  the  price  paid  for  one, 

X 

72 
and  =  the  price  if  there  had  been  6  more. 

x  +  6  ^ 

By  the  conditions,  —  =  — ^^  +  1. 

^  X      x  +  6 

Solvmg,  a:  =  18  or  -  24. 

Only  the  positive  value  of  x  is  admissible,  as  the  negative  value 
does  not  answer  to  the  conditions  of  the  problem.  The  number  of 
sheep,  therefore,  was  18. 

Note  1.  In  solving  problems  which  involve  quadratics,  there  will 
always  be  two  values  of  the  unknown  quantity ;  but  only  those  values 
should  be  retained  as  answers  which  satisfy  the  conditions  of  the  prob- 
lem. 


224  ALGEBRA. 

Note  2.  If,  in  the  given  problem,  the  words  "  6  more  *'  had  been 
changed  to  "  6  fewer"  and  "  $  1  apiece  less  "  to  "  $  1  apiece  more"  we 
should  have  found  the  answer  24. 

In  many  cases  where  the  solution  of  a  problem  gives  a  negative 
result,  the  wording  may  be  changed  so  as  to  form  an  analogous  prob- 
lem to  which  the  absolute  value  of  the  negative  result  is  an  answer. 


PROBLEMS. 

3.  I  bought  a  lot  of  flour  for  $175  ;  and  the  number  ot 
dollars  per  barrel  was  \  of  the  number  of  barrels.  How 
many  barrels  were  purchased,  and  at  what  price  ? 

4.  Separate  the  number  15  into  two  parts  the  sum  of 
whose  squares  shall  be  117. 

6.   Find  two  numbers  whose  product  is  126,  and  quotient 

6.  I  have  a  rectangular  field  of  corn  containing  6250 
hills.  The  number  of  hills  in  the  length  exceeds  the  num- 
ber in  the  breadth  by  75.  How  many  hills  are  there  in  the 
length,  and  in  the  breadth? 

7.  Find  two  numbers  whose  difference  is  9,  and  whose 
sum  multiplied  by  the  greater  is  266. 

8.  The  sum  of  the  squares  of  two  consecutive  numbers  is 
113.     What  are  the  numbers? 

9.  A  man  cut  two  piles  of  wood,  whose  united  contents 
were  26  cords,  for  $35.60.  The  labor  on  each  cost  as  many 
dimes  per  cord  as  there  were  cords  in  the  pile.  Required 
the  number  of  cords  in  each  pile. 

10.  Find  two  numbers  whose  sum  is  8,  and  the  sum  of 
whose  cubes  is  152.    -• 

11.  Find  three  consecutive  numbers  su-ch  that  twice  the 
product  of  the  first  and  third  is  equal  to  the  square  of  the 
second  increased  by  62. 


PROBLEMS.  225 

12.  A  grazier  bought  a  certain  number  of  oxen  for  $240. 
Having  lost  3,  he  sold  the  remainder  at  $8  a  head  more  than 
they  cost  him,  and  gained  $59.     How  many  did  he  buy? 

13.  A  merchant  bought  a  quantity  of  flour  for  $96.  If 
he  had  bought  8  barrels  more  for  the  same  money,  he  would 
have  paid  $2  less  per  barrel.  How  many  barrels  did  he 
buy,  and  at  what  price  ? 

14.  Find  two  numbers,  whose  product  is  78,  such  that  if 
one  be  divided  by  the  other  the  quotient  is  2,  and  the 
remainder  1. 

15.  The  plate  of  a  rectangular  looking-glass  is  18  inches 
by  12.  It  is  to  be  framed  with  a  frame  all  parts  of  which 
are  of  the  same  width,  and  whose  area  is  equal  to  that  of 
the  glass.     Required  the  width  of  the  frame. 

16.  A  merchant  sold  a  quantity  of  flour  for  $39,  and 
gained  as  much  per  cent  as  the  flour  cost  him.  What  was 
the  cost  of  the  flour? 

17.  A  certain  company  agreed  to  build  a  vessel  for 
$6300;  but,  two  of  their  number  having  died,  the  rest  had 
each  to  advance  $  200  more  than  they  otherwise  would  have 
done.  Of  how  many  persons  did  the  company  consist  at 
first? 

18.  Divide  the  number  24  into  two  parts,  such  that  the 
sum  of  the  fractions  obtained  by  dividing  24  by  them  shall 
be||. 

19.  A  detachment  from  an  army  was  marching  in  regular 
column,  with  6  men  more  in  depth  than  in  front.  When  the 
enemy  came  in  sight,  the  front  was  increased  by  870  men, 
and  the  whole  was  thus  drawn  up  in  4  lines.  Required  the 
number  of  men. 

20.  A  merchant  sold  goods  for  $16,  and  lost  as  much  per 
cent  as  the  goods  cost  him.  What  was  the  cost  of  the 
goods  ? 


226  ALGEBRA. 

21.  A  certain  farm  is  a  rectangle,  whose  length  is  twice 
its  breadth.  If  it  should  be  enlarged  20  rods  in  length,  and 
24  rods  in  breadth,  its  area  would  be  doubled.  Of  how 
many  acres  does  the  farm  consist? 

22.  A  square  court-yard  has  a  gravel-walk  around  it. 
The  side  of  the  court  lacks  one  yard  of  being  6  times  the 
breadth  of  the  walk,  and  the  number  of  square  yards  in  the 
walk  exceeds  the  number  of  yards  in  the  perimeter  of  the 
court  by  340.  Find  the  area  of  the  court  and  the  width  of 
the  walk. 

23.  A  merchant  bought  54  bushels  of  wheat,  and  a  cer- 
tain quantity  of  barley.  For  the  former  he  gave  half  as 
many  dimes  per  bushel  as  there  were  bushels  of  barley,  and 
for  the  latter  40  cents  a  bushel  less.  He  sold  the  mixture 
at  $1  per  bushel,  and  lost  $57.60  b}^  the  operation.  Re- 
quired the  quantity  of  barley,  and  its  price  per  bushel. 

24.  A  certain  number  consists  of  two  digits,  the  left- 
hand  digit  being  twice  the  right-hand.  If  the  digits  are 
inverted,  the  product  of  the  number  thus  formed,  increased 
by  11,  and  the  original  number,  is  4956.     Find  the  number. 

25.  A  cistern  can  be  filled  by  two  pipes  running  together 
in  2  hours  55  minutes.  The  larger  pipe  by  itself  will  fill  it 
sooner  than  the  smaller  by  2  hours.  What  time  will  each 
pipe  separately  take  to  fill  it? 

26.  A  and  B  gained  by  trade  $1800.  A's  money  was  in 
the  firm  12  months,  and  he  received  for  his  principal  and 
gain  $2600.  B's  money,  which  was  $3000,  was  in  the  firm 
16  months.     How  much  money  did  A  put  into  the  firm? 

27.  My  gross  income  is  $1000.  After  deducting  a  per- 
centage for  income  tax,  and  then  a  percentage,  less  by  one 
than  that  of  the  income  tax,  from  the  remainder,  the  income 
is  reduced  to  $912.  Find  the  rate  per  cent  of  the  income 
tax. 


PROBLEMS.  227 

28.  A  man  travelled  102  miles.  If  he  had  gone  3  miles 
more  an  hour,  he  would  have  performed  the  journey  in  5| 
hours  less  time.     How  many  miles  an  hour  did  he  go? 

29.  The  number  of  square  inches  in  the  surface  of  a 
cubical  block  exceeds  the  number  of  inches  in  the  sum  of  its 
edges  by  210.     What  is  its  volume? 

30.  A  man  has  two  square  lots  of  unequal  size,  contain- 
ing together  15,025  square  feet.  If  the  lots  were  contigu- 
ous, it  would  require  530  feet  of  fence  to  embrace  them  in  a 
single  enclosure  of  six  sides.  Required  the  area  of  each 
lot. 

31.  A  set  out  from  C  towards  D  at  the  rate  of  3  miles  an 
hour.  After  he  had  gone  28  miles,  B  set  out  from  D  towards 
C,  and  went  every  hour  -^^  of  the  entire  distance  ;  and  after 
he  had  travelled  as  many  hours  as  he  went  miles  in  an  hour, 
he  met  A.     Required  the  distance  from  C  to  D. 

32.  A  courier  proceeds  from  P  to  Q  in  14  hours.  A  sec- 
ond courier  starts  at  the  same  time  from  a  place  10  miles 
behind  P,  and  arrives  at  Q  at  the  same  time  as  the  first 
courier.  The  second  courier  finds  that  he  takes  half  an  hour 
less  than  the  first  to  accomplish  20  miles.  Find  the  dis- 
tance from  P  to  Q. 

33.  A  person  bought  a  number  of  $20  mining-shares 
when  they  were  at  a  certain  rate  per  cent  discount  for 
$  1500  ;  and  afterwards,  when  they  were  at  the  same  rate  per 
cent  premium,  sold  them  all  but  60  for  $1000.  How  many 
did  he  buy,  and  what  did  he  give  for  each  of  them? 


228  ALGEBRA. 


XXIII.  EQUATIONS  IN  THE  QUADRATIC 
FORM. 

269.  An  equation  is  in  the  quadratic  form  when  it  is 
expressed  in  three  terms,  two  of  which  contain  the  unknown 
quantity ;  and  of  these  two,  one  has  an  exponent  twice  as 
great  as  the  other;  as, 

a^  _j_  ipl  =  72  ; 
(a;2-i)2_}-3(a^_l)  =  18;  etc. 

270.  The  rules  for  the  solution  of  quadratics  are  applica- 
ble to  equations  having  the  same  form. 

1 .  Solve  the  equation  ic^  —  6  a^  =  1 6 . 

Completing  the  square, 

fl;6_6a;3  4-9  =  16  +  9  =  25. 

Extracting  the  square  root, 

a^-3  =  ±5. 

Whence,  a^  =  3±5  =  8or  —  2. 

Extracting  the  cube  root,   ic=  2  or  — -^2,  Ans. 

Note.  There  are  also  four  imaginary  roots,  which  may  be  obtained 
by  the  method  explained  in  Art.  282. 

2.  Solve  the  equation  2x-h3-y/x=27. 

Since  ^x  is  the  same  as  x^,  this  is  in  the  quadratic  form. 
Multiplying  by  8,  and  adding  3^  or  9  to  both  members, 
16a;  +  24Va'  +  9  =  216  +  9  =  225. 

Extracting  the  square  root, 

.4Va^  +  3  =  ±15. 
Or,  4Va;  =  -3±15  =  12or  -18, 


QUADRATIC  EQUATIONS.  229 

9 
Whence,  V^  =  3  or  —  -  • 

81 
Squaring,  a;  =  9  or  — ,  Ans, 

3.   Solve  the  equation  16 a;" ^  —  22  ic" ^  =  3. 
Multiplying  by  16,  and  adding  11^  to  both  members, 

le^a;"^  -  16  X  22a;~*  +  121  =  48  +  121  =  169. 
Extracting  the  square  root, 

16aj"*-ll  =  ±13. 
Or,  16a;"^  =  11  ±  13  =  -  2  or  24. 

Whence,  a;~*  =  — -  or  -• 

'  8         2 

Extracting  the  cube  root, 


Raising  to  the  fourth  power. 


1       X        /'3\f 
-1=  JL  or  '     ^^ 
16 

Inverting  both  members,    a;=  16  or  ( -  j  ,  Ans, 

V 

Note.  In  solving  equations  of  the  form  xi  =  a,  first  extract  the 
root  corresponding  to  the  numerator,  and  afterwards  raise  to  the  power 
corresponding  to  the  denominator.  Particular  attention  should  be  paid 
to  the  algebraic  signs ;  see  Arts  192  and  201. 

EXAMPLES. 
Solve  the  following  equations  : 

4.  a;*-25ic2  =  -144.  7.    aj--*- 9a;-2  =  - 20. 

5.  »«+20iB8-^69  =  0.  8.    81a^4--  =  82. 

6.  a;i«  +  31ar''-32  =  0,     '  9.    8x^-216  =  373:^. 


230  ALGEBKA. 

10.  (3a^-2)2-ll(3a;2_2)  +  l0  =  0. 

11.  (a^-5)2  =  241-29a^. 

12.  a^--a;^  =  56.  17.    2x~'-j-61x~^ -96=^0, 

13.  x^-{-x^=766.  18.   4x-15  =  17VaJ. 

14.  2/.+  3.-«-56  =  0.  19.   ^g±2=4_i. 

^  4+V^       V^ 

15.  3 aj^  + 0^^  =  3104.  20.    3a5^-^'=- 592. 

16.  3(c^4-26a;^  =  -16.  21.    8aj~^- 15a;~^- 2  =  0. 

271.  An  equation  may  be  solved  with  reference  to  an 
expression,  by  regarding  it  as  a  single  quantity. 

1.    Solve  the  equation  (x  —  5y  —  3{x  —  5)^  =z 40. 

Regarding  x  —  5  as  a  single  quantity,  we  complete  the 
square  in  the  usual  way.  Multiplying  by  4,  and  adding  9  to 
both  members, 

4(a;  -  5)3  -  12(a;  -  5)*  +  9  =  160  +  9  =  169. 
Extracting  the  square  root, 

2(a;-5)^-3  =  ±13. 
Or,  2(a;-5)^  =  3±  13  =  16  or  -10. 

Whence,  (a;  — 5)^  =  8  or  -5. 

Extracting  the  cube  root, 

(a;-5)^  =  2or --^5. 
Squaring,  a?  —  5  =  4  or  ^25. 

Transposing,  a;=  9  or  5  +-^25,  Ans. 


QUADRATIC   EQUATIONS.  231 

An  equation  of  the  fourth  degree  may  sometimes  be 
solved  by  expressing  it  in  the  quadratic  form. 

2.    Solve  the  equation  a;* -|- 12 ar^ 4- 343^- 12 a;- 35  =  0. 
We  may  write  the  equation  as  follows  : 

(a;^+12aT^  +  36a^)-2aj2-12a;  =  35. 
Or,  ^  (a^  +  6a;)^-2(a^+6a;)  =  35. 

Completing  the  square, 

Extracting  the  square  root,      (a:^  +  6a;)  —  1  =  ±  6. 

Whence,  a?-\'ex=l±6=    7  or—  5. 

Completing  the  square,  »^  +  6  a;  +  9  =  16  or  4. 

Extracting  the  square  root, 

x  +  3=  ±4or  ±2. 

Whence,  a;=  -3  ±  4  or  -3  ±  2 

=  1,  —7,  —1,  or  —5,  Ans, 

Note.  In  solving  equations  like  the  above,  the  first  step  is  to  form 
a  perfect  square  with  the  .r*  and  x^  terms,  and  a  portion  of  the  x^ 
term.  By  Art.  261,  the  third  term  of  the  square  is  the  square  of  the 
quotient  obtained  by  dividing  the  x^  term  by  twice  the  square  root  of 
the  x*  term. 


3.   Solve  the  equation  2a;2^.  V2ar'  +  1  =  H. 
Adding  1  to  both  members, 

(2ar^  + 1)  +  V2^Tl  =  12. 
Completing  the  square, 


4(2a;2  +  l)4.4V2ar^-f  1  +  1  =  48  +  1=49. 
Extracting  the  square  root, 

2  V2FT1  +  1  =  ±  7. 


Or,  2V2a:2  4.i  =  _i±7  =  6or-8. 


Whence,  \^2x^-{-l  =  3  or  - 4, 


232  ALGEBRA. 

Squaring,  2ic^  4- 1  =  9  or  16. 

2a^  =  8  or  15. 

a^  =  4or— . 
2 

Extracting  the  square  root,  a;  =  ±  2  or  ±  -  V^O,  Ans, 

Note.  In  solving  equations  of  this  form,  add  such  quantities  to 
both  members  that  the  expression  without  the  radical  in  the  first  mem- 
ber may  be  the  same  as  that  within,  or  some  multiple  of  it. 


EXAMPLES. 
Solve  the  following  equations  : 

5.   aJ*+10a^+17i»2-- 40a; -84  =  0. 


6.   ic2 -  10a; -2VV-^10^+18+ 15  =  0. 


7.   a;2_j_5+Va;2_j.5^i2. 


8.  2a;2+3a;-5V2^N-3¥+9  =  -3. 

9.  fl;4  +  2iB5-25a;2_26a;  +  120  =  0. 
10.   iB^-6a;3-29a;2_,_ii4a,^30. 


11.  a2_6a._|.5Va;2_g^^20  =  46. 

12.  Va;+10-</aj+10=2. 


13.  4a;2_,_ey4a;2^i2a;-2  =  -3-12a?. 

14.  (a;3^i6)f_3(aj3_|.  16)^  +  2  =  0. 

15.  4(aj-l)^-5(a;-l)^  +  l  =  0. 

16.  a;*+14a;3  +  47a;2_i4^_43^Q^ 


17.  3(ar'  +  5a;)-2V^M-5^Tl=2. 

18.  (a;-a)^  +  2V&(a;-a)^-36  =  0. 


SIMULTANEOUS  EQUATIONS.  283 

XXIV.  SIMULTANEOUS  EQUATIONS. 

INVOLVING  QUADRATICS. 

272.  The  degree  of  an  equation  containing  more  than  one 
unknown  quantity  is  determined  by  the  greatest  sum  of  the 
exponents  of  the  unknown  quantities  in  any  term.     Thus, 

2x-^3xy  =  4:     is  an  equation  of  the  second  degree. 

ic2  _  fj^y2^  __  ^53  jg  g^jj  equation  of  the  third  degree. 

Note.  This  definition  assumes  that  the  equation  has  been  cleared 
of  fractions,  and  freed  from  radical  signs  and  fractional  and  negative 
exponents. 

273.  Two  equations  of  the  second  degree  with  two  un- 
known quantities  will  generally  produce,  by  elimination,  an 
equation  of  the  fourth  degree  with  one  unknown  quantity. 
The  rules  for  quadratics  are,  therefore,  not  sufficient  to 
solve  all  simultaneous  equations  of  the  second  degree. 

In  several  cases,  however,  the  solution  may  be  effected  by 
the  ordinary  rules. 

Case  I. 

274.  When  one  equation  ia  of  the  first  degree. 

Equations  of  this  kind  may  always  be  solved  by  finding 
the  value  of  one  of  the  unknown  quantities  in  terms  of  the 
other  from  the  simple  equation,  and  substituting  the  result 
in  the  other  equation. 

1 .    Solve  the  equations    |  2  aj^  -  a?y  =  6  y.  (1 ) 

1     x^2y=l.  (2) 

From  (2),  2y  =  7-x,ory  =  ^l^=-^'  (3) 

Substituting  in  (1) ,  2ar^  -  ^{~^^  =  6  f^^=^\ 


234 


ALGEBRA. 


Clearing  of  fractions,  4a^  —  7ic  +  a^  =  42  —  6a;. 
Or,  5a^-x==^2. 


Solving  this  equation. 
Substituting  in  (3) , 


a;  =  3  or  — 


14 


7+ii 

.,      7-3         ^5 


=  2  or 


2  2 

49 


10 


Ans,  a;  =  3,  2/  =  2  ;  or,  a?  =  -  i^,  2/  =  — • 

5  10 


EXAMPLES. 
Solve  the  following  equations  : 


Xsx  +    y  =1. 

3^    (a;  +  2/  =  ~l. 
1x2/  =  — 56. 

4    1^5  -2/  =      3. 

(ar^ +2^^=117. 

g^    {10x  +  y  =  Sxy. 
\      x  —  y=:  —  2. 

6.  ja;'-2/^  =  -37. 
la;  —y  =—    1. 

7,  (a;-2/=5. 
la;2/  =  -6. 

ja;  +2/  = 


9.  ^ 


2  +  3 

=  4. 

?+?  = 

La;      2/ 

:1. 

10.   I  «^  + 2/^  =  152. 
la;  +^  =     2. 

j^j     |3a;2_2a;2/=15. 
l2a;  +32/   =12. 

12.   j8a^-2/^  =  -7. 
i2a;  -2/  =-1. 

j3     (•a;2^3^^_^2 
la;  +2w=7. 


=  23. 


8. 


=    8. 

29. 


14. 


+  22/ 
^-1-^  =  5. 

y    X     2* 

3a;  — 22/=  — 4. 
/ 


SIMULTANEOUS  EQUATIONS.  285 

Case  II. 

275.  When  the  equations  are  symmetrical  with  respect  to  x 
and  y. 

Note  1.  An  equation  is  symmetrical  with  respect  to  two  quantities 
when  they  can  be  interchanged  without  destroying  the  equality. 

Thus,  x^  —  ary  +  y^  =  3  is  symmetrical,  for  on  interchanging  x  and  y 
it  becomes  y^  —  yx  ■\-  x^  =  3,  which  is  equivalent  to  the  first  equation. 
But  a:  —  y  =  1  is  not  symmetrical,  for  on  interchanging  x  and  y  it 
becomes  y  —x=  1,  which  is  a  different  equation. 

In  solving  equations  by  the  symmetrical  method,  they 
must  be  combined  in  such  a  way  as  to  give  the  values  of  the 
sum  and  difference  of  the  unknown  quantities. 

iic  +  V  =  2.  n^ 

an/ =  -15.  (2) 

Squaring  (1),  ar^-f  2a!y-f  3/^  =  4.  (3) 

Multiplying  (2)  by  4,  4a^  =  -  60*.  (4) 

Subtracting  (4)  from  (3),    a^- 2ict/4-/ =  64. 
Extracting  the  square  root,  x  —  y=z±S.  (5) 

Adding  (1)  and  (5),  2a;  =  2  ±  8  =  10  or  -  6. 

Whence,  x=    5  or  —  3. 

Subtracting  (5)  from  (1) ,       2?/  =  2  q:  8  =  -  6  or  10. 
Whence,  y  =  — 3or5. 

Ans.  x  =  5,  y  =  —  S;  or,  a;  =  —  3,  y  =  6. 

Note  2.  The  signs  ±  and  if  before  two  quantities  signify  that  when 
the  first  quantity  is  +,  the  second  is  — ;  and  when  the  first  is  — ,  the 
second  is  +.  Thus,  in  the  above  solution,  when  2ar  =  2  +  8,  2y  =  2— 8; 
and  when  2a:  =  2-8,  2y  =  2  +  S.  That  is,  when  x  =  5,  y  =  -S;  and 
when  x=  —S,  y  =  6. 

In  the  operation,  the  sign  db  is  changed  to  rp  whenever  +  would  be 
changed  to  — . 

Note  3.  The  above  equations  may  also  be  solved  as  in  Case  I. ;  but 
the  symmetrical  method  is  shorter,  and  more  elegant. 


236  ALGEBRA. 

«     ^  ,                     .  ra;2_|-2/2  =  50.               (1) 

2.  Solve  the  equations  < 

^  \x  -y  =    8,               (2) 

Squaring  (2),  a^ -2xy -{-y^  =  M,               (3) 

Subtracting  (3)  from  (1),  2xy  =  -U.           (4) 

Adding  (1)  and  (4),  a^ +  2xy +  y^  =  S6, 

Whence,  x  +  y  =  ±Q,             (5) 

Adding  (2)  and  (5),  2a;  =  8  ±  6  =  14  or  2. 

"Whence,  x=    7  or  1 . 

Subtracting  (2)  from  (5) ,  22/  =  -  8  ±  6  =  -  2  or  -  14. 

"Whence,  ^  =  —  1  or  —  7. 
Ans.  a;=7,  ?/  =  — 1;  or,  a;=l,   y  —  —  l. 

Note  4.  The  symmetrical  method  may  often  be  used  in  cases  like 
the  above,  where  the  equations  are  symmetrical  except  in  the  signs  of 
the  terms. 

«     o  ,        ,               .  (           0^  +  2/3=133.              (1) 

3.  Solve  the  equations  \    ^                                         ^ 

'  \x'-xy-\-y''=    19.              (2) 

Dividing  (1 )  by  (2) ,  x  +  y=l.                 (3) 

Squaring,  a^  +  2  a;?/  +  ?/-  =  49.               (4) 

Subtracting  (4)  from  (2) ,  —^xy=z  —  30. 

Or,  -xy  =  - 10.           (5) 

Adding  (2)  and  (5),  x^ -2xy +  y^  =  d. 

Whence,  x  —  y  =  ±S,             (6) 

Adding  (3)  and  (6),  2a;=  7±3  =10  or  4. 

Whence,  a;  =  5  or  2. 

Subtracting  (6)  from  (3),  2^/=  7:f  3  =4  or  10. 

Whence,  ?/  =  2  or  5. 

Ans,  a;  =  5,  2/  =  2  ;  or,  a?  =  2,  2/  =  5, 


SIMULTANEOUS  EQUATIONS.  287 


EXAMPLES. 

Solve  the  following  equations : 

4     (a;  +  2/=l. 
'  \xy=-6. 

12.  ^-^=2* 

Lxy  =  eo, 

(x-y=    6. 

13.  1^  +  2/^=^5. 
liB2/  =  42. 

6.    f^-2/  =  -10. 
(a^  =  -21. 

14.  |^-2/'  =  -316. 
*    (a;  —  2/  =—     4. 

•  Xa^-^xy  +  y'=ld, 

16    l^  +  /=193. 
•  \x  +2/  =-5. 

g     (0^  +  2/^  =  25. 
*    U2/=12. 

16.  |«'  +  2/  =  12. 
(a^  =  — 45. 

la^  +  2/2  =  58. 

17     (a:^-2/«  =  -65. 
•  \x^-\-xy  +  f=13, 

10.  1^-2/^=98. 
U  -2/  =2. 

18.  l^-^+^^i^y. 

(a;  —2/  =  —  5. 

11     fa^  +  2/«  =  9. 
U  +2/  =3. 

19     fa^H-2r^  =  ~386. 
•  la;  +y  =-     2. 

Case  IH. 

276.  When  each  equation  is  of  the  second  degree,  and 
homogeneous  (Art.  35). 

Note.  Certain  examples,  in  which  the  equations  are  of  the  second 
degree  and  homogeneous,  may  be  solved  by  the  method  of  Case  II. 
The  method  of  Case  III.  should  be  used  only  when  the  equations  can 
be  solved  in  no  other  way. 


238  ALGEBRA. 

1.    Solve  the  equations  <    „ 

^  U2-f2/2^29. 

Putting  y=vxm  the  given  equations,  we  have 

k'-2vx'=    5;or,a^  =  — - —  (1) 

'  l-2v  ^  ^ 

iB2+7;2a^  =  29;  or,  a^         ^^ 


Equating  the  values  of  a^. 


1  + 
5  29 


1-2-?;      1+^2 
5  4-5^2=29-58^. 

5^2^.58^  =  24. 

2 
Solving  this  equation,  v  =  -  or  —  12. 

5 

^                                                             5  5 

Substituting  these  values  in  (1 ) ,  a^  = or 


1-1        1+24 
5 

=  25  or  i. 


Whence,  aj  =  ±  5  or  ± 

Substituting  the  values  of  v  and  x  in  the  equation  y  =  'ya;, 


3/  =  |(±5)or-12('±^^=±2orT 


12 

V5" 


^ws.    a;  =±5,        2/=i2; 

1  12 

or,  a;=±-V5,  2/  =  ?:---V^' 
o  5 


Note.   In  finding  y  from  the  equation  y  =  vx,  care  must  be  taken  to 
multiply  each  pair  of  values  ef  x  by  the  corresponding  value  of  v. 


SIMULTANEOUS  EQUATIONS.  289 


EXAMPLES. 
Solve  the  following  equations'i 

'   Xxy  +  f  =18,  _  •  Xxy-f=    2. 

3     (2x^-{-xy  =  15.  ^     „  ('2/-4aj?/  +  3a^  =  17. 

(x'j^xy-y'^-n.  g  {2x^-2xy-f=    3. 

lar^  +  2/2=i3.  *  (    a^4-3a^  +  /  =  ll. 

|^  +  a:2/  +  42/2  =  6.  g  |  6a^-5a;2/  + 2/=  12. 

*   l3ic2  +  8/=14.  *  l3a;2_,.2a;y_32,2==_3. 


"11 


icy  —  xr  =  5. 
lSx'-31xy-\-16f=z2i, 


Solve  the  equations   ■< 


MISCELLANEOUS  EXAMPLES. 

277.  No  general  rules  can  be  given  for  the  solution  of 
examples  which  do  not  come  under  the  cases  just  consid- 
ered. Various  artifices  are  employed,  familiarity  with  which 
can  only  be  obtained  by  experience. 

a^-f=19,  (1) 

x'y-xy^=    6.  (2) 

Multiplying  (2)  by  3,    Sx^y-3xy^=  18.  (8) 

Subtracting  (3)  from  (1), 

x^ —  Sa^y -\-3xy^  —  y^=    1. 
Extracting  the  cube  root,        x  —  y=l.  (4) 

Dividing  (2)  by  (4),  xy=    6.  (5) 

Equations  (4)  and  (5)  may  now  be  solved  by  the  method 
of  Case  II.     We  shall  find  a;  =  3  or  —  2,  and  i/  =  2  or  —  3. 
Ans.  ic=3,  2/  =  2;  or,  a;  =  —  2,  t/  =  — 3. 


240  ALGEBRA. 


X 


a^-^y^=9  xy. 
+  y  =6. 


Putting  x=^u-\-v  and  y  =  u  —  v.,  we  have 

(u-\-vy+{u-vy=9(u+v){u-v).    (1) 
(u  +  v)  +{u-v)  =6.  (2) 

Reducing  (2),  2w=6,  orw  =  3. 

Reducing  (1),  2u^-\-6uv^  =  9  {u^ ~v^). 

Substituting  the  value  of  w, 

54  +  18^2^9  (9 --y^). 
Whence,  'v^=l,or'i;  =  ±l. 

Therefore,  a;  =  ?*  +  v  =  3±l  =  4or2, 

y  =  u  —  v  =  3^:l  =  2  or  4:. 

Ans.  a;  =  4,  2/  =  2  ;  or,  a;  =  2,  ?/  =  4. 

Note.   The  artifice  of  substituting  u  +  u  and  u  —  v  for  x  and  i/  is 
advantageous  in  any  case  where  the  given  equations  are  symmetrical, 

3.    Solve  the  equations 

cx^  +  f-{-2x-{-2y  =  2S.  (1) 

I  xy=    6.  (2) 

Multiplying  (2)  by  2,  2xy=  12.  (8) 

Adding  (1)  and  (3), 

a^  +  20^2/ +  2/2 -f  2a;  +  22/ =  35. 
Or,  (a; +  2/)' +  2  (a; +  2/)  =35, 

Completing  the  square, 

(a;  +  2/)^  +  2(aj  +  2/)4-l=36. 
Whence,  (a;  +  2/)  +  1  =  ±  6, 

a;  4-  2/  =  —  1  ±  6  =  5  or  —  7.  (4) 

Squaring  (4),  0^2  + 2aJ2/  +  2/^  =  25  or  49.  (5) 

Multiplying  (2)  by  4,  4.xy  =  2A.  (6) 

Subtracting  (6)  from  (5), 

a^  —  2xy  +  y^=l  or  25. 


SIMULTANEOUS  EQUATIONS.  241 

Whence,  x  —  y=±l  or  ±5.  (7) 

Adding  (4)  and  (7),  2x=  5  ±  1  or  -7  ±  5, 

ic  =  3,  2, -1,  or  —  6. 
Subtracting  (7)  from  (4),  2y  =  5zfl,  or  -7:f5, 

2/ =  2,  3,  -6,  or  -1. 
Ans.  x  =  3,  y  =  2;  x=2,  y  =  S; 

a:  =  -l,2/  =  -6;  or,  aj  =  -  6,  y  = -1. 


fx*  +  y^=    97. 
4.    Solve  the  equations       < 

ix  +y  =-1. 

Putting  x^u-\-v  and  y  =  u  —  v,  we  have, 

{u-\-vy-\-{u-vy=    97.  (1) 

(u  +  v)  -^{u-v)  =-1.  (2) 

Keducing  (2) ,  2  it  =  —  1 ,  or  w  =  —  -• 

Keducing  (1),     2u*4- 12uV  +  2 v*=    97. 

Substituting  the  value  of  w, 

Ij^Sv^-\-2v*=    97. 
8 


Solving  this  equation. 

'i;2  = 

f . "  - 

31 
'  4* 

Whence, 

1?  = 

■■±1   or 

^V-31 
2 

Therefore, 

x  =  u-{-v 

= 

1 
2 

*i»- 

2           2 

-31 

y  =  u-'V 

= 

2  or 

1 
2 

—  3  or 

-1±V^ 
2 

2^       2 

-31 

» 

id 

-31 

== 

-3 

or  2  or 

-i^V- 

2 

-31 

r 

1 

242 


ALGEBRA. 


EXAMPLES. 
Solve  the  following  equations  : 


f,     (xy  —  2x=      5. 

•  \xy-\-Sy  =  -2. 

*  {■^x+-^y  =  3. 

y     C    4a^-3/  =  -ll. 
llla^  +  5?/2=  301. 

g     C    a;3+   2/^  =  35. 
la^y  +  a;/  =  30. 


9.  ^ 


2^      a?      10 


3a7-2i/  =  4. 

\  x  —y  =  m. 

11,    {x^-\-f  +  x-hy=18. 
\xy=z6. 

j2     (a^-2x2/  =  16. 
•   \2xy  +  f==-3. 

13.   K  +  2/ 

la;  +2/ 


2/^=  18a;2/. 
12. 


x^  +  3xy=—U. 
xy-^Ay^=      30. 


16. 


16 


^1+1=.  11. 

X      y 


1 

13^2/ 


=  18. 


(  x  —  y=  a^b. 
\xy=:2a^  +  2ab. 

17     ('a;''  +  2/^  =  9-a;. 
\a^-y^=e. 


18. 
20 


'1  J.  1       ari 

1-1=11. 

'  x^J^y^-x-^y  = 

:18 

xy  +  x-{-y  = 

19, 

*  J  ar^  +  iC2/    +  2/^  =      7. 


rflj22^H-a;2/^  =  6. 
21.  jl      1^2 

,2.   1-^  +  2/^=17. 
U  -y  =   3. 

23.  I'^^-^/^-^a'. 
(  X  —2/  =a. 


Divide  tlie  second  equation  by  the  first. 


24 


SIMULTANEOUS  EQUATIONS.  243 

^y^  +  y  =    19. 


(x+x^y^  +  y  =    \^.  3j  (  a;  +  2/  =  3(a-6). 

•  \o?-^xy     +2/2=133.  *  \xy=2o?-bah-\-2h^. 

2g     ra;  +  22/  =  3a4-&.  32  |ar'  +  ^  =  33. 

•  Uy  +  2/'=2a(a  +  6).  *  U +1/  =   3. 

26     fa^2/+^/=   30.  33  ixf+y^l, 

'   la^2/'  +  a^2/'  =  ^68.  '  («22,*  +  2/'  =  5. 

•  la^(a;-y)=2a26-263.  34.  i2y-Sz  =  -5. 

(.ar2  +  /-22=ll. 

^        ^  Uicy-ar^    +62/2=44. 

f^  +  lV^  =  2§.  fa;-v      a;  +  2/_3 

^^-  J2/       V2/       4  36.  J  ^T^     ^-^"2' 

^^-y  =  ^'  [2a^-2/  =  7. 

30     (a^-a52^4-^  =  19.  g^  (  x' +  f=7 -{-xy. 

'   l2x'-y^  =  -17.  '  Xa^-[-f  =  6xy-l, 


PROBLEMS. 

278.  Note.  In  the  following  problems,  as  in  those  of  Chap.  XXII., 
only  those  answers  are  to  be  retained  which  satisfy  the  conditions  of 
the  problem. 

1.  The  sum  of  the  squares  of  two  numbers  is  106,  and 
the  difference  of  their  squares  is  -^  the  square  of  their  differ- 
ence.    Find  the  numbers. 

2.  What  two  numbers  are  those  whose  difference  multi- 
plied by  the  less  produces  42,  and  by  their  sum,  133? 

3.  The  sum  of  the  areas  of  two  square  fields  is  1300 
square  rods,  and  it  requires  200  rods  of  fence  to  enclose 
both.     What  are  the  areas  of  the  fields  ? 


244  ALGEBRA. 

4.  The  difference  of  the  squares  of  two  numbers  is  7, 
and  the  product  of  their  squares  is  144.     Find  the  numbers. 

5.  If  the  length  of  a  rectangular  field  were  increased  by 
2  rods,  and  its  breadth  by  3  rods,  its  area  would  be  108 
square  rods ;  and  if  its  length  were  diminished  by  2  rods, 
and  its  breadth  by  3  rods,  its  area  would  be  24  square  rods. 
Find  the  length  and  breadth  of  the  field. 

6.  The  sum  of  the  cubes  of  two  numbers  is  407,  and  the 
sum  of  their  squares  exceeds  their  product  by  37.  Find 
the  numbers. 

7.  A  man  bought  6  ducks  and  2  turkeys  for  $15.  He 
bought  four  more  ducks  for  $14  than  turkeys  for  $9. 
What  was  the  price  of  each  ? 

8.  Find  a  number  of  two  figm-es,  such  that  if  its  digits 
are  inverted,  the  sum  of  the  number  thus  formed,  and  the 
original  number,  is  33,  and  their  product  is  252. 

9.  The  sum  of  the  terms  of  a  fraction  is  8.  If  1  is 
added  to  each  term,  the  product  of  the  resulting  fraction 
and  the  original  fraction  is  |.     Required  the  fraction. 

10.  A  rectangular  garden  is  surrounded  by  a  walk  7  feet 
wide ;  the  area  of  the  garden  is  15,000  square  feet,  and  of 
the  walk  3696  square  feet.  Find  the  length  and  breadth 
of  the  garden. 

11.  A  rectangular  field  contains  160  square  rods.  If  its 
length  be  increased  by  4  rods,  and  its  breadth  by  3  rods,  its 
area  is  increased  by  100  square  rods.  Find  the  length  and 
breadth  of  the  field. 

12.  A  man  rows  down  stream  12  miles  in  4  hours  less 
time  than  it  takes  him  to  return.  Should  he  row  at  twice 
his  ordinary  rate,  his  rate  down  stream  would  be  10  miles  an 
hour.  Find  his  rate  in  still  water,  and  the  rate  of  the 
stream. 


SIMULTANEOUS   EQUATIOIS^S.  245 

13.  A  and  B  bought  a  farm  of  104  acres,  for  which  they 
paid  $320  each.  On  dividing  the  land,  A  said  to  B,  "If 
you  will  let  me  have  my  portion  in  the  situation  which  I 
shall  choose,  you  shall  have  so  much  more  land  than  I,  that 
mine  shall  cost  $3  an  acre  more  than  yours."  B  accepted 
the  proposal.  How  much  did  each  have,  and  at  what  price 
per  acre? 

14.  If  the  product  of  two  numbers  be  added  to  their  sum, 
the  result  is  47  ;  and  the  sum  of  their  squares  exceeds  their 
sum  by  62.     Find  the  numbers. 

Note.  Let  the  numbers  be  represented  hy  x  -\-  y  and  x  —  y. 

15.  The  sum  of  two  numbers  is  7,  and  the  sum  of  their 
fourth  powers  is  641.     Required  the  numbers. 

16.  The  fore- wheel  of  a  carriage  makes  15  more  revolu- 
tions than  the  hind-wheel  in  going  180  yards ;  but  if  the 
circumference  of  each  wheel  were  increased  by  3  feet,  the 
fore-wheel  would  make  only  9  more  revolutions  than  the 
hind-wheel  in  the  same  distance.  Find  the  circumference 
of  each  wheel. 

17.  A  man  has  $1300,  which  he  divides  into  two  por- 
tions, and  loans  at  different  rates  of  interest,  so  that  the  two 
portions  produce  equal  returns.  If  the  first  portion  had 
been  loaned  at  the  second  rate,  it  would  have  produced  $36  ; 
and  if  the  second  portion  had  been  loaned  at  the  first  rate, 
it  would  have  produced  $49.     Required  the  rates  of  interest. 

18.  Cloth,  when  wetted,  shrinks  \  in  its  length  and  Jg-  in 
its  width.  If,  the  surface  of  a  piece  of  cloth  is  diminished 
by  5f  square  yards,  and  the  length  of  the  four  sides  by  4^ 
yards,  what  were  the  length  and  width  of  the  cloth  origi- 
nally ? 


246  ALGEBRA. 


XXV.  THEORY  OF  QUADRATIC  EQUA- 
TIONS. 

279.    Denoting  the  roots  of  the  equation  a^  -i-px  =  qhy  r^ 
and  ra,  we  have  (Art.  267), 

n  =  -P  +  ^P'  +  JQ^  and  r,  =  -P-^p'  +  ig. 


Adding  these  values, 

—  2» 
^i+?'2  =  — ^  =  -i?. 

Multiplying  them  together, 

nr,  =  -P'-(P^  +  ^g)  (Art.  95)  ^=^  =  -q. 

That  is,  if  a  quadratic  equation  he  reduced  to  the  form 
x^  -{-px  =  q,  the  algebraic  sum  of  the  roots  is  equal  to  the 
coefficient  of  x  with  its  sign  changed^  and  the  product  of  the 
roots  is  equal  to  the  second  member^  with  its  sign  changed. 

Example,  Required  the  sum  and  product  of  the  roots  of 
the  equation  2a^  —  7a;— 15  =  0. 

The  equation  may  be  written  in  the  form 

2        2 

7  15 

Whence,  the  sum  of  the  roots  is  -,  and  their  product  is  — — • 

2  2 

280.  The  principles  of  Art.  279  may  be  used  to  form  a 
quadratic  equation  which  shall  have  any  required  roots. 

For,  denoting  the  roots  of  the  equation  x^  +px  —  q=:0  by 
fj  and  rg,  we  have,  by  the  preceding  article, 

P  =  -{ri-\- r^) ,  and  -  5'  =  r^r^. 


THEORY  OF   QUADRATIC   EQUATIONS.         247 
We  may  therefore  write  the  equation  in  the  form 

or,  a:^  —  TiX  —  r^-^  r{i\  —  0. 

That  is  (Art.  105) ,  (a;  —  r^  {x  —  r^  =  0. 

Hence,  to  form  an  equation  which  shall  have  any  required 
roots, 

Subtract  each  of  the  roots  from  x,  and  place  the  product  of 
the  resulting  expressions  equal  to  zero. 

7 
Example.     Form  the  equation  whose  roots  are  4  and . 

By  the  rule,  {x  —  A)fx-\-^z=  0. 

Multiplying  by  4,      (a;  —  4)  (4a;  +  7)  =  0. 

Or,  4ar'-9a;-28  =  0,  Ans. 

EXAMPLES. 

281.  Find  by  inspection  the  sum  and  product  of  the  roots 
of: 

1.  a;2-f-5a;  +  2  =  0.  5.    8a^-a;-h4  =  0. 

2.  af-7x-i-n  =  0.  6.   6a;-4ar'  +  3  =  0. 

3.  a^-\-6x-l  =  0.  7.    7-12a;-14a;2^Q^ 

4.  2ic2-3a;-2  =  0.  8.    4ar^-4aa;  +  a^-6^  =  0. 
Form  the  equations  whose  roots  are : 


9.  4,  5.           11.  3,  -?. 

0 

13.  ?,  I 
3' 4 

"•-!■-? 

10.  1,  -3.       12.  7,  -i^. 
3 

14.   -?,i 
3    7 

ie.-l!,.. 

17.    a-6,  a  +  26. 

19.    2+V3 

,  2-V3. 

18.    m(l+m),m(l-m).      20.   !2i±V»^  mW?. 

^  2 


248  ALGEBRA. 

282.  By  Art.  280,  the  equation  x^-\-px  —  q  =  0  may  be 
written  in  the  form  (x  —  Vi)  (a;  —  j-g)  =  0,  where  rj  and  rg  are 
its  roots. 

It  will  be  observed  that  the  roots  may  be  obtained  by 
placing  the  factors  of  the  first  member  separately  equal  to 
zero^  and  solving  the  simple  equations  thus  formed. 

This  principle  is  often  used  in  solving  equations : 

1.  Solve  the  equation  (2aj -  3)  (3.t  +  5)  =  0. 

Placing  the  factors  separately  equal  to  zero, 

2aj-3  =  0,  or  a;  =  5; 
2 

and  3a;  +  5  =  0,  or  a;  = 

o 

Ans.  ic  =  -,  or . 

2  3 

2.  Solve  the  equation  aj^  —  5a^  —  24ic  =  0. 
Factoring  the  first  member,  x{x  —  d>){x-\-^)  =  0. 
Therefore,  cc  =  0  ; 

flj  — 8  =  0,  ora;  =  8; 
and  a;H-3  =  0,  or  a;  =  — 3. 

Ans.  ic  =  0,  8,  or  —3. 

3.  Solve  the  equation  a:^4-4aj2  —  a;  —  4  =  0. 
Factoring  the  first  member  (Art.  105), 

(a;  +  4)(a.'2-l)  =  0. 
Therefore,  a;  +  4  =  0,  ora;  =  — 4; 

and  a^  — 1  =  0,  oraj=±l. 

Ans,  aj  =  —  4  or  ±  1 , 


THEORY  OF  QUADRATIC  EQUATIONS.        249 

4.    Solve  the  equation  x^  —  1  =0. 

Factoring  the  first  member, 

(.T-l)(ar  +  a;-j-l)  =  0. 
Therefore,  a?  —  1  =  0,  or  a;  =  1 ; 

and  a^  +  .T+l  =  0.  (1) 

2 


Solving  (1)  by  the  rules  for  quadratics,  x  = 


Ans.  a;  =  1  or 

EXAMPLES. 
Solve  the  following  equations  : 

5.  /'a;_?Ya;  +  I'j  =  0.  10.    2ic3-18a;  =  0. 

6.  ^x'-x^O.  11.  (3a;  +  l)(4a^-25)  =  0. 

7.  {ax  +  h){hx-a)  =  0,  12.  3a^  +  12a.-2  =  0 

8.  (a^-4)(aj2-9)  =  0.  13.  {x'-a'')(x'-ax-h)=0, 

9.  (a;-2)(x24-9a;+20)=0.  14.  "14.^ -^x" ^I2x  =  0. 

15.  ic(2a;  +  5)(3a;-7)(4a;+l)  =  0. 

16.  (a^ -  5a;  +  6)  (a^  +  7a;  +  12)  {x'-^x -  4)  =  0. 

17.  a?-\-\=:Q,  19.   a.-3-a;2-9a;  +  9  =  0. 

18.  a;«-l=0.  20.    2a;3  +  3a;2_2a;_3  =  0. 

Note.  The  above  examples  are  illustrations  of  the  important  prin- 
ciple that,  the  degree  of  an  equation  indicates  the  number  of  its  roots ; 
thus,  an  equation  of  the  third  degree  has  three  roots ;  of  the  fourth 
degree,  four  roots ;  etc. 

It  should  be  observed  that  the  roots  are  not  necessarily  unequal; 
thus,  the  equation  x^  — 2x  +  l  =  0  may  be  written  (x  — l)(a:  — 1)  =  0, 
and  therefore  the  two  roots  are  1  and  1. 


250  ALGEBRA. 

FACTORING. 

283.  A  quadratic  expression  is  a  trinomial  expression  of 
the  form  ay?-\-hx-\- c. 

Any  such  expression  may  be  resolved  into  two  simple  fac- 
tors by  the  artifice  of  completing  the  square  (Art.  260),  in 
connection  with  Art.  111. 

EXAMPLES. 
1.    Factor  6a;2  +  7a;-3. 


6ar^+7a;-3=6(a^  + 


7^_1\ 
6       2/ 

By  Art.  262,  the  expression  in  the  parenthesis  will  become 

a  perfect  square  if  the  third  term  is  (  —  j  ;  hence, 

"'^— [-+¥-(fj-(fj-i] 

=  (2a;-}-3)(3ic-l),  Ans, 
2.  Factor  4 a^  — 20a; +  19. 
4 ar^ -  20a;  +  19  =  4a;2  _  20a;  +  25  -  25  +  19 
=  (2a; -5)2- 6 
=  (2a;-'5+V6)(2^-5-V6)i  ^ns. 


THEORY   OF   QUADRATIC   EQUATIONS.         251 


Note.    If  the  x^  term  is  negativey  the  entire  expression  should  be  en- 
closed in  a  parenthesis  preceded  by  a  —  sign. 


3.    Factor  102  +  11a;  -  a.-^. 


102) 


:(-^)"-f] 


102 


=  -    X 


i'-'i^m 


11      2S\ 

2        2  J 


=  (x-{-6)-{-l){x-17) 
=  (6+ a;)  (17  — a;),  Ans. 
4.    Factor  ar^  — a;2/  — 21/^  — 5  a; -I-2/-I- 6. 
a^  —  xy  —  5x  —  2y^-\-y-\-Q 


i'-'-^) 


y^+lOyH-25      8y^ 
4 


4?/ -24 


_/^     .V  +  5V     9y^  +  6y  +  l 


(-f') 


=    a;- 


.V  +  5 


+^X'"-4^'-^)  (^^*-"^' 


=  (a;  +  2/  — 2)(a;  — 2?/  — 3),  ^ws. 
Factor  the  following : 

5.  ft^_4a;_60. 

6.  a^  +  13a;  +  40. 

7.  x'-Ux  +  lS, 

8.  2a.-2-7a;-15. 

9.  4a^-15a;  +  9. 


10.  5x'-}-S6x-j-7. 

11.  39-10a;-a:2^ 

12.  2  +  X-6X', 

13.  a;2  +  4a;  +  l. 

14.  9a;2-6a;-4. 


252  ALGEBRA. 

15.  Sx^-lSx  +  d,  22.  1-Sx-af. 

16.  6-x-2ay'.  23.  16 -\- 26x-24.a^. 

17.  5-\-4:X-12x^.  24.  25a;2-20ic-2. 

18.  9a^-12a;+l.  25.  6a;2- 13aa;- ISa^. 

19.  5-18ir-8a^.  26.  20 a^ -i- 41  mx  +  20 m\ 

20.  10a;2-23aj  +  6.  27.  12 x^ -\- 7 xy  -  10 y\ 

21.  16a^  +  34a;  +  15.  28.  21a:2_  53^^^^  21 7^^. 

29.  x'^  +  xy -Qy^-\-x-{-lSy-6. 

30.  a^  +  3rc2/  +  2?/-  +  3.T  +  4?/4-2. 

31.  Q-5y-j-x-6y^-\-6xy-x^. 

32.  aj^  — 5 x?/4- 62/^  —  0  372;  +  142/2  +  40^. 

33.  2a^-i»2/-7/  +  3aj+3?/-2. 

34.  3a2  +  4a6  +  62  +  5a-6-12. 

Certain  expressions  of  the  fourth  degree  may  be  resolved 
into  two  quadratic  factors  by  the  artifice  of  completing  the 
square. 

35.  Factor  a*  +  a^^s  4- 54. 

By  Art.  108,  the  expression  will  become  a  perfect  square 
if  the  middle  term  is  2a^b^.     Hence, 

a*  +  a'b^  +  6^  =  (a*  +  2  a'b^  +  b')  -  a'b^ 
=  (a' -\- by  -  a'b' 

=  (a^  -j-b^-j-  ah)  {a"  +  6^  _  ^j)      (^^.^^  m) 
=  (^2  4.  a6  +  62)  (^2  _ab  +  b') ,  ^ns. 

36.  Factor  9a;^- 39  a^  + 25. 

9  ic^  -  39  a^  +  25  =  (9  o;^  -  30a^  +  25)  -  9 0.-2 
=  {Sx^-5y-9x' 
=  (Sx^  -  6  +  Sx)  {Sx''  -  5  -  Sx) 
=  (3a;2  +  3a;-5)(3a;2-3a;-5),  Ans, 


THEORY  OF  QUADRATIC  EQUATIONS.         25S 

37.  Factor  a;* +1. 

=  (x^  +  iy-(x^2y 

Factor  the  following : 

38.  x^  +  x'-hl.  46.  a*-5aV  +  a;\ 

39.  a;^-7a^  +  l.  47.  ic^  +  81. 

40.  4a*-8a262  +  6^  48.  4.a* -\- 15  a'b^-^Ub'. 

41.  m^-14?MV  +  n^  49.  16a:^- 49mV4-9m*. 

42.  l-1362  +  46\  50.  9a;*-6a;2-f-4. 

43.  X*- 12x^7/ -\-4y\  61.  9a*4- 14aW  +  25m*. 

44.  4a*  +  Sa^  +  9.  52.  4-32n2  +  49n*. 

45.  4m^-24?/i2+25.  53.  16 x*  -  49  x^f  +  25 y\ 

284.  The  equation  a;*-f- 1  =0  may  be  solved,  as  in  Art. 
282,  by  placing  the  factors  of  the  first  member  (Ex.  37,  Art. 
283)  separately  equal  to  zero  ;  thus, 

-v/2  +  a/ 2 

a^  +  a;V2  +  l  =  0;  whence  a;  =  — YJl±^ f; 

and      7?  —  X  -y/2  -|-  1  ==  0  ;  whence  x  =  -  ~ 


Therefore, 


2 

±V2±V^^ 


EXAMPLES. 
Solve  the  following  equations  : 

1.  x*-\-16  =  0.  4.    x'-{-a*  =  0. 

2.  »^-6a;2 4-1  =  0.  5.    0^^-8.^2  +  4  =  0. 

3.  x'-x'  +  l  =  0.  6.    a;*-  — +  1  =  0. 


254  ALGEBRA. 

DISCUSSION  OF  THE  GENERAL  EQUATION. 
285.    The  roots  of  the  equation  a;-  -\-px=  q  are 


'2  '2 

We  will  now  discuss  these  values  for  different  values  of  p 
and  q. 

I.  Suppose  q  positive. 

Since  p^  is  essentially  positive  (Art.  192),  the  quantity 
under  the  radical  sign  is  positive  and  greater  than  p^. 

Therefore  the  value  of  the  radical  is  greater  than  p. 

Hence  Vi  is  positive  and  rg  is  negative. 

If  p  is  positive,  rg  is  numerically  greater  than  r^ ;  that  is, 
the  negative  root  is  numerically  the  greater. 

If  p  is  2;ero,  the  roots  are  numerically  equal. 

If  p  is  negative,  r^  is  numerically  greater  than  r.2 ;  that  is, 
the  positive  root  is  numerically  the  greater. 

II.  Suppose  q=0. 

The  quantity  under  the  radical  sign  is  now  equal  to  p^,  so 
that  the  value  of  the  radical  is  p. 

If  p  is  positive,  ?'i=  0,  and  r^  is  negative. 
If  p  is  negative,  ri  is  positive,  and  r.^  =  0. 

III.  Suppose  q  negative,  and  4g  numerically  <p^. 

The  quantity  under  the  radical  sign  is  now  positive  and 
less  than  p^. 

Therefore  the  value  of  the  radical  is  less  than  p. 
If  p  is  positive,  both  roots  are  negative. 
If  p  is  negative,  both  roots  are  positive. 

IV.  Suppose  q  negative,  and  ^q  numerically  =  p^. 

The  quantity  under  the  radical  sign  is  now  equal  to  zero. 
Therefore  the  roots  are  equal ;  being  negative  if  p  is  posi- 
tive, and  positive  it  p  is  negative. 


THEORY  OF  QUADRATIC  EQUATIONS.    255 

V.    Suppose  q  negative,  and  4tq  numerically  >p^. 

The  quantity  under  the  radical  sign  is  now  negative ; 
hence,  by  Art.  201,  both  roots  are  imaginary. 

The  roots  are  both  rational  or  both  irrational  according  as 
p^  +  4g  is  or  is  not  sl  perfect  square. 

EXAMPLES. 

1.  Determine  by  inspection  the  nature  of  the  roots  of  the 

equation  2 a;^  —  5a;  —  18  =  0. 

5x 
The  equation  may  be  written  x^ =  9. 

Since  q  is  positive  and  p  negative,  the  roots  are  one  posi- 
tive and  the  other  negative ;  and  the  positive  root  is  numeri- 
cally the  greater. 

25  169 

In  this  case,  p^  +  4^= }-  36  = ;  a  perfect  square. 

4  4 

Hence  the  roots  are  both  rational. 

Determine  by  inspection  the  nature  of  the  roots  of  the  fol- 
lowing : 

2.  a;2_,_2a;_i5  =  o.  6.    6a^-7ar-5  =  0. 

3.  ic2  +  5a;-f  6  =  0.  7.    9ar  + 30a;  =  -  25. 

4.  ic2-10a;  =  -25.  8.    9ar^-f 8  =  18a;. 

6.    3x'2-5a;-f-4  =  0.  9.    10  -  3a;  -  18a;2^  0^ 


2b6  ALGEBRA. 


XXVI.    INEQUALITIES. 

2586.  An  Inequality  is  a  statement  that  one  of  two  quanti- 
ties is  greater  or  less  than  the  other ;  as, 

a  >  &,  or  m  <  n. 

The  terms  greater  and  less  are  here  taken  in  the  algebraic 
sense ;  that  is,  of  any  two  quantities  a  and  6,  a  is  the  greater 
when  a  —  6  is  positive,  and  the  less  when  a  —  &  is  negative. 

287.  The  expression  on  the  left  of  the  sign  of  inequality 
is  called  the  First  Member,  and  that  on  the  right  the  /Second 
Member,  of  the  inequality. 

288.  Two  inequalities  are  said  to  subsist  in  the  same  sense 
when  the  first  member  is  the  greater  or  the  less  in  each. 

Thus, 

a  >  6,  and  c>d;  or  m  <  n,  and  p<q, 

are  inequalities  which  subsist  in  the  same  sense. 

289.  Two  inequalities  are  said  to  subsist  in  a  contrary 
sense  when  the  first  member  is  the  greater  in  one,  and  the 
less  in  the  other. 

Thus,  a  >  6,  and  c  <  d 

are  inequalities  which  subsist  in  a  contrary  sense. 

290.  An  inequality  will  continue  in  the  same  sense  after  the 
same  quantity  has  been  added  to,  or  subtracted  from,  both 
members. 

For  consider  the  inequality  a^b. 
Then  by  Art.  286,  a  —  6  is  positive. 
Therefore  each  of  the  quantities 

(a-^c)  —  (b  +  c),  and  (a  —  c)  —  (b  —  c) 

is  positive,  since  each  is  equal  to  a  —  b. 


INEQUALITIES.  257 

Whence  by  Art.  286, 

a  +  c  >  6  -h  c,  and  a  —  c>h  —  c. 

It  follows  from  the  above  that  a  term  may  he  transposed 
from  one  member  of  an  inequality  to  the  other  by  changing  its 
sign. 

291.  If  the  signs  of  all  the  terms  of  an  inequality  are 
changed,  the  sign  of  inequality  must  be  reversed. 

For  consider  the  inequality 

a  —  b>c  —  d. 
Transposing  each  term  (Art.  290) ,  we  have 

d  —  c>b  —  a. 
That  is,  b  —a<d-c. 

292.  An  inequality  will  continue  in  the  same  sense  after 
both  members  have  beeri  multiplied  or  divided  by  the  same 
positive  quantity. 

For  consider  the  inequality  a>b. 

By  Art.  286,  a  —  6  is  positive. 

Hence,  if  m  is  positive,  each  of  the  quantities 

,x         ,  a  —  b 


m 

or, 

ma  —  mb,    and 7 

m      m 

is  positive. 

ma  >  m6,    and  —  >  — 
'           mm 

Therefore, 

293.  If  both  members  of  an  inequality  are  multiplied  or 
divided  by  the  same  negative  quantity,  the  sign  of  inequality 
must  be  reversed. 

For  multiplying  or  dividing  by  a  negative  quantity  changes 
the  signs  of  all  the  terms,  and  hence  the  sign  of  inequahty 
must  be  reversed  (Art.  291). 


258  ALGEBRA. 

294.  If  any  number  of  inequalities^  subsisting  in  the  same 
sense,  are  added  member  to  member,  the  resulting  inequality 
will  also  subsist  in  the  same  sense. 

For  consider  the  inequalities 

a>b,  a'>b',  a">b",  .... 

Then  each  of  the  quantities  a  —  b,  a'  —  b',  a"  —  b", ...,  is 
positive. 

Therefore  their  sum 

a-b  +  a'-b'+a"-b"-\-'", 

or,  a  +  a'+  a"+ {b  +  6'+  &"+  •••) 

is  positive. 

Whence,    a  +  a'+  c^"+  •••> 5  +  &'+&"  +  •••. 

Note.  If  two  inequaUties,  subsisting  in  the  same  sense,  are  sub- 
tracted member  from  member,  the  resulting  inequality  will  not  neces. 
sarily  subsist  in  the  same  sense. 

Thus,  if  a  >  6  and  a'  >  6',  then  a  — -  6  and  a'  —  b'  are  positive. 

But  a  —  b  —  (a'  —  b'),  or  its  equal,  a  —  a'  —  (b  —  b'),  may  be  eithei 
positive,  negative,  or  zero ;  and  hence  it  does  not  necessarily  follow 
that  a  — a'>6  — 6'. 

EXAMPLES. 

295.  1.  Find  the  limit  of  x  in  the  inequality 

„        23  ^2a;  ,   . 

7x < \-5, 

3        3 

Clearing  of  fractions  (Art.  292),  we  have 

21a;— 23  <  2a; +15. 

Transposing  (Art.  290) ,  and  uniting  terms, 

19a;<38. 

Whence  by  Art.  292,    a;<  2,  Ans. 


INEQUALITIES.  259 

2.    Find  the  limits  of  x  and  y  in  the  following  : 

|3a;  +  2y>37  (I) 

(2a;  +  3y  =  33  (2) 

Multiplying  (1)  by  3,  and  (2)  by  2, 
9a;  +  62/>lll 

Subtracting,  5  a;  >  45 

Whence,  a;>9. 

Multiplying  (1)  by  2,  and  (2)  by  3, 

6aj  +  42/>74 

6a;  +  9y=99 

Subtracting,  —  5  ?/  >  —  25 

Whence  by  Art.  293,        y<h. 

Therefore,  a;  >  9,  and  2/  <  5,  Ans, 

Find  the  limits  of  x  in  the  following : 

3.  (6a;  +  l)2-105<(4a;-3)(9a;  +  4). 

4.  (2a;4-3)(3a;-l)>(2a;  +  7)(3x-2)+l. 

5.  (x  +  l)(a;  +  2)(a;-3)>(a;-l)(a;-4)(a;  +  5). 

6.  3aa;  +  14a6>  Ga^H- 76a;,  if  3  a  —  76  is  negative. 

7.  ^~^  <  ^~    ,  if  a  and  6  are  positive  and  a  >  6. 

h  a 

Find  the  limits  of  x  and  y  in  the  following : 
g     (5a;4-72/>38.  g    |2a;  +  32/<57. 

1    a;-    y  =  -2.  *   l3a;  +  72/  =  93. 

10.  Find  the  limits  of  x  when 

2a;— 9>  21  — 4a;,  and  3  a;  — 11  >  5a;  — 41. 

11.  A  certain  positive  whole  number,  plus  23,  is  less  than 
6  times  the  number,  minus  12  ;  and  9  times  the  number, 
minus  54,  is  less  than  twice  the  number,  plus  9.  What  is 
the  number? 


260  ALGEBRA. 

12.  A  teacher  being  asked  the  number  of  his  pupils,  re- 
plied that  29  was  less  than  twice  their  number,  diminished 
by  7;  and  that  5  times  their  number,  diminished  by  5,  was 
less  than  twice  their  number,  increased  by  55.  Required  the 
number  of  his  pupils. 

13.  A  shepherd  has  a  number  of  sheep  such  that  twice 
the  number,  diminished  b}*  45,  exceeds  79,  diminished  by 
twice  the  number;  and  5  times  the  number,  increased  by  1, 
is  less  than  3  times  the  number,  increased  by  69.  How 
many  sheep  has  he  ? 

14.  Prove  that  if  a  and  b  are  positive, 

0      a 
Since  the  square  of  any  quantity  is  positive, 

(a-by>0. 
That  is,  a2_2a6+62>o, 

or,  a'-{-b^>2ab. 

Dividing  each  term  of  the  inequality  by  ab  (Art.  292) ,  we 
have 

b     a 

15.  Prove  that  for  any  value  of  a;,  .'^  —  3  a?  +  4  >  If. 

16.  Prove  that  for  any  values  of  a  and  6, 

(2a  +  6)(2a-&)>26(6a-5  6). 

17.  Prove  that  for  any  values  of  a,  6,  and  c, 

a^  +  62  +  c^  >  2  {ab  -\-bc-  ca) . 

18.  Prove  that  (a^  -  b^)  (c^  -  d')  <  (ac  -  bdy. 

J9.   Prove  that  if  a^  +  6^ ^  1  and  c^  +  d'^=l,  then 


THE   THEORY  OF   LIMITS.  261 


XXVII.    THE    THEORY    OF    LIMITS. 

INTERPRETATION   OF   THE   FORMS   -»   -.   AND   -• 

0     <»  0 

Note.    The  symbol  oo  is  called  Infinity. 

296.  A  variable  quantity^  or  simply  a  variable^  is  a  quan- 
tity which  may  assume,  under  the  conditions  imposed  upon 
it,  an  indefinitely  great  number  of  different  values. 

A  constant  is  a  quantity  which  remains  unchanged  through- 
out the  same  discussion. 

297.  A  limit  of  a  varinble  is  a  constant  quantity,  the  dif- 
ference between  whicii  and  the  variable  may  be  made  less 
than  any  assigned  quantity  however  small,  without  ever 
becoming  zero. 

In  other  words,  a  limit  of  a  variable  is  a  fixed  quantit}'  to 
which  the  variable  approaches  indefinitely  near,  but  never 
actually  reaches. 

The  variable  is  said  to  approach  indefinitely  to  its  limit. 

298.  Suppose,  for  example,  that  a  point  moves  from  A 
towards  B  under  the  condition  that  it 

shall    move,   during    successive  equal     , ,         i     i    i 

intervals  of  time,  first  from  A  to  O, 

half-way  between  A  and  B ;   then  to  D,  half-way  between 

C  and  B  ;  then  to  E,  half-way  between  D  and  B  ;  and  so  on 

indefinitely. 

In  this  case  the  distance  between  the  moving  point  and  B 
can  be  made  less  than  any  assigned  quantity  however  small, 
but  cannot  be  made  equal  to  zero. 

Hence  the  distance  from  A  to  the  moving  point  is  a  vari- 
able which  approaches  indefinitely  the  constant  value  AB  as 
a  limit,  without  ever  reaching  it. 

Again,  the  distance  from  the  moving  point  to  5  is  a  vari- 
able which  approaches  the  limit  0. 


262  ALGEBRA. 

299.    The  Theorem  of  Limits.    If  two  variables  are  always 
equal,  and  each  approaches  a  limit,  the  tiuo  limits  are  equal. 


A 

1 

M 

t 

C 

1 

B 

1 

A' 

1 

M' 

\ 

B> 

.,,1 

Let  AM  and  AM'  be  two  equal  variables  which  approach 
the  limits  AB  and  A'B',  respectively. 

If  possible,  suppose  AB  >  A'B\  and  lay  off  AC=  A'B'. 

Then  the  variable  AM  may  assume  values  between  AC 
and  AB,  while  the  variable  A'M'  is  restricted  to  values  less 
than  AC',  which  is  contrary  to  the  hypothesis  that  the  vari- 
ables should  always  be  equal. 

Hence  AB  cannot  be  >  A'B',  and  in  like  manner  it  may 
be  proved  that  AB  cannot  be  <A'B' ;  therefore  AB=A'B'. 


INTERPRETATION   OF 
300.    Consider  the  series  of  fractions 


a     a       a         a 
3'    y   ^'    ^003' 


where  each  denominator  is  one-tenth  of  the  preceding 
denominator. 

It  is  evident  that,  by  sufficiently  continuing  the  series,  the 
denominator  may  be  made  less  than  any  assigned  quantity 
liowever  small,  and  the  value  of  the  fraction  may  be  made 
greater  than  any  assigned  quantity  however  great. 

In  other  words, 

If  the  7iumerator  of  a  fi-action  remains  constant,  while  the 
denominator  approaches  the  limit  0,  the  value  of  the  fraction 
increases  without  limit. 

It  is  customary  to  express  this  principle  as  follows : 
a 


THE   THEORY   OF   LIIVUTS.  263 


INTERPRETATION    OF  ±. 

oo 

301.  Consider  the  series  of  fractions 

a      a_       a         a 

3'   30'    300'    3000'  ***' 

where  each  denominator  is  ten  times  the  preceding  denomi- 
nator. 

It  is  evident  that,  by  sufficiently  continuing  the  series,  the 
denominator  may  be  made  greater  than  an}'  assigned  quan- 
tity however  great,  while  the  value  of  the  fraction  may  be 
made  less  than  any  assigned  quantity  however  small. 

In  other  words, 

If  the  numerator  of  a  fraction  remains  constant,  while  the 
denominator  increases  without  limits  the  value  of  the  fraction 
approaches  the  limit  0. 

It  is  customary  to  express  this  principle  as  follows : 

«=0. 

00 

302.  The  student  must  understand  clearly  that  no  absolute 
meaning  can  be  attached  to  such  results  as 

-  =  oc,  or  —  =  0  ; 

for  there  can  be  no  such  thing  as  division  unless  the  divisor 
and  quotient  are  Jinite  quantities. 

If  such  forms  occur  in  mathematical  investigations,  they 
must  be  interpreted  as  indicated  in  Arts.  300  and  301.  (Com- 
pare the  Note  to  Art.  405.) 


THE  PROBLEM  OF  THE  COURIERS. 
303.    The  discussion  of  the  following  problem  will  serve 

to  further  illustrate  the  form  -,  besides  furnishing  an  inter- 

0 
pretation  of  the  form  — 


264  ALGEBRA. 

Two  couriers,  A  and  B,  are  travelling  along  the  same  road 
in  the  same  direction,  ER',  at  the  rates  of  m  and  n  miles  an 
hour  respectively.  If  at  any  time,  sa}^  12  o'clock,  A  is  at  P, 
and  B  is  a  miles  from  him  at  Q,  at  what  time  and  at  what 
point  are  they  together? 

B  P  Q  Rt 

I I \ I 

Let  X  =  the  required  time  in  hours  after  12  o'clock, 

and  y  =  the  distance  travelled  by  A  in  the  time  x,  or 

the  distance  in  miles  from  F  to  the  point 

of  meeting. 
Then  y  —  a  =  the  distance  travelled  by  B  in  the  time  x^  or 

the  distance  in  miles  from  Q  to  the  point 

of  meeting. 

Since  the  distance  equals  the  rate  multiplied  by  the  time, 

I         y=mx, 
\y  —  a  =  nx. 

Solving  these  equations,  we  obtain 

a 

X  — 1 

m  —  n 

am 

y  = 

m  —  n 

We  will  now  discuss  these  values  under  different  hy- 
potheses. 

1.    m>7i. 

In  this  case  the  values  of  x  and  y  are  positive. 

Hence  the  couriers  are  together  at  some  time  after  12 
o'clock,  and  at  some  point  to  the  right  of  P. 

This  corresponds  with  the  supposition  made  ;  for  if  m  is 
greater  than  n,  A  is  travelling  faster  than  J5,  and  it  is  evi- 
dent that  he  will  eventually  overtake  him  at  some  point  in 
advance  of  their  positions  at  12  o'clock. 


THE  PROBLEM  OF  THE   COURIERS.  -265 

2.    m<n. 

In  this  case  the  vahies  of  x  and  y  are  negative. 

Hence  the  couriers  are  together  at  some  time  before  12 
o'clock,  and  at  some  point  to  the  left  of  P.  (Compare 
Art.  44.) 

This  corresponds  with  the  hypothesis  ;  for  if  7n  is  less  than 
n,  A  is  travelling  more  slowly  than  B,  and  they  must  have 
been  together  before  12  o'clock,  and  before  they  could  have 
advanced  as  far  as  P. 

3.    m  =  w,  or  m  —  n  =  0. 
In  this  case  the  values  of  x  and  y  take  the  forms 

a        ,  am 
-,  and  — , 

respectively. 

As  m  —  71  approaches  the  limit  0,  the  values  of  x  and  y 
increase  without  limit  (Art.  300)  ;  hence,  if  m  =  m,  no  finite 
values  can  be  assigned  to  x  and  y,  and  the  problem  is  im- 
possible. 

This  interpretation  corresponds  with  the  supposition  made  ; 
for  if  m  is  equal  to  7i,  the  couriers  are  a  miles  apart  at  12 
o'clock,  and  are  travelling  at  the  same  rate ;  and  it  is  evi- 
dent that  they  never  could  have  been,  and  never  will  be 
together. 

Thus,  an  infinite  result  indicates  that  the  problem  is  im- 
possible. 

4.    a  =  0,  and  m  >  w  or  m<.  n. 

In  this  case  we  have  x  =  0  and  y  =  0. 

Hence  the  couriers  are  together  at  12  o'clock,  at  the 
point  P. 

This  corresponds  with  the  hypothesis ;  for  if  a  =  0,  and 
m  and  n  are  unequal,  the  couriers  are  together  at  12  o'clock, 
and  are  travelling  at  unequal  rates ;  hence  they  never  could 
have  been  together  before  that  time,  and  they  never  will  be 
together  afterwards. 


266  ALGEBRA. 

5.    a  =  0,  and  m  =  n.  "      ' 

In  this  case  both  x  and  y  take  the  form  — 

According  to  the  supposition  made,  the  couriers  are 
together  at  12  o'clock,  and  are  travelling  at  the  same  rate. 

Therefore  they  always  must  have  been,  and  always  will 
be  together. 

There  is  in  this  case  no  single  answer  nor  finite  number  of 
answers  to  the  problem  ;  for  any  value  of  x  whatever,  to- 
gether with  the  corresponding  value  of  y^  will  satisfy  the 
given  conditions. 

Hence,  a  result  -  indicates  that  the  problem  is  indetermi- 
0 

nate;  that  is,  the  number  of  solutions  is  indefinitely  great. 


RATIO   AND  PROPORTION.  267 


XXVIII.    RATIO  AND    PROPORTION. 

304.  Ratio  is  the  relation  with  respect  to  magnitude 
which  one  quantity  bears  to  another  of  the  same  kind,  and  is 
expressed  by  writing  the  first  quantity  as  the  numerator  and 
the  second  as  the  denominator  of  a  fraction. 

Thus  the  ratio  of  a  to  6  is  -  ;  and  it  is  also  expressed  a:b. 

b 

305.  A  Proportion  is  an  equality  of  ratios. 

Thus,  if  the  ratio  of  a  to  6  is  equal  to  the  ratio  of  c  to  d, 
they  form  a  proportion,  which  may  be  written  in  either  of  the 
forms:  ^      ^ 

a:b  =  c:d,    -  =  -,     or    a:b::c:d. 
b      d 

306.  The  first  term  of  a  ratio  is  called  the  antecedent^  and 
the  second  term  the  consequent. 

Thus  in  the  ratio  a :  6,  a  is  the  antecedent,  and  b  is  the 
consequent. 

The  first  and  fourth  terms  of  a  proportion  are  called  the 
extremes^  and  the  second  and  third  terms  the  means. 

Thus  in  the  proportion  a:b  =  c:d,  a  and  d  are  the  ex- 
tremes, and  b  and  c  the  means. 

307.  In  a  proportion  in  which  the  means  are  equal,  either 
mean  is  called  a  Mean  Proportional  between  the  first  and 
last  terms,  and  the  last  term  is  called  a  Third  Proportional 
to  the  first  and  second  terms. 

A  Fourth  Proportional  to  three  quantities  is  the  fourth 
term  of  a  proportion  whose  first  three  terms  are  the  three 
quantities  taken  in  their  order. 

Thus  in  the  proportion  a:b  =  b:c^  b  is  a  mean  propor- 
tional between  a  and  c,  and  c  is  a  third  proportional  to  a 
and  b. 

In  the  proportion  a  :  6  =  c  :  d,  d  is  a  fourth  proportional  to 
c:,  6,  and  c. 


268  ALGEBRA. 

308.  A  Continued  Proportion  is  a  series  of  equal  ratios, 
in  which  each  consequent  is  the  same  as  the  following  ante- 
cedent; as, 

a  :  b  =^  b  :  c  =  c  :  d  =  d  :  e. 

PROPERTIES  OF  PROPORTIONS. 

309.  In  any  proportion  the  product  of  the  extremes  is  equal 
to  the  product  of  the  means. 

Let  the  proportion  be     a  :  5  =  c  :  d. 

Then  by  Art.  305,  ^  =  -- 

•^  b      d 

Clearing  of  fractions,        ad  =  be. 

310.  A  mean  proportional  between  two  quantities  is  equal 
to  the  square  root  of  their  product. 

Let  the  proportion  he     a  :  b  =  b  :  c. 
Then  by  Art.  309,  b^=ac. 

Whence,  b  =  Vac. 

311.  From  the  equation  ad  =  bc,  we  obtain 

be        J    ,      ad 
a  =  — ,   and   b=  — 
d  c 

That  is,  in  any  proportion  either  extreme  is  equal  to  the 

product  of   the  means  divided  by  the  other  extreme ;    and 

either  mean  is  equal  to  the  product  of  the  extremes  divided 

by  the  other  mean. 

312.  (Converseof  Art.  309.)  If  the  product  of  two  quantities 
is  equal  to  the  product  of  two  others^  one  pair  may  be  made 
the  extremes,  and  the  other  the  means ^  of  a  proportion. 

Let  ad  =  be. 

^.  .,.      ,     ,  ,  ad      be         a     c 

D.v.d.ngbyM,  ^  =  ^'"'1=1 

Whence,  a:b  =  c:d. 


RATIO   AND  PROPORTION.  269 

In  a  similar  manner  we  ma}^  prove  that : 
a  :  c  =  6  :  d, 
b  :  d  =  a  :  c, 
c:d  =  a:b,  etc. 

313.  In  any  proportion  the  terms  are  in  proportion  by 
Alternation  ;  that  is,  the  first  term  is  to  the  third,  as  the  second 
term  is  to  the  fourth. 

Let  a  :  b  =  c  :  d. 

Then  by  Art.  309,  ad  =  be. 

Whence  by  Art.  312,      a:c  =  b:d, 

314.  In  any  proportion  the  terms  are  in  proportion  by 
Inversion ;  that  is,  the  second  term  is  to  the  first,  as  the  fourth 
term  is  to  the  third. 

Let  a  :  b  =  c  :  d. 

Then,  ad  =  be. 

Whence,  b  :  a  =  d  :  c. 

315.  In  any  proportion  the  terms  are  in  proportion  by 
Composition ;  that  is,  the  sum  of  the  first  two  terms  is  to  the 
first  term,  as  the  sum  of  the  last  two  terms  is  to  the  third  term. 

Let  a  :  b  =  c:  d. 

Then,  ad=:bc. 

Adding  both  members  to  ac, 

ac-{-ad  =  ac  +  be, 
or,  a(c  +  d)  =  c(a-\-b). 

Whence  (Art.  312), 

a-f6:a  =  c  +  d:c. 
Similarly  we  may  prove  that 

a  +  b:b  =  c-\-d'.d. 


270  ALGEBRA. 

316.  In  any  proportion  the  terms  are  in  proportion  by  Divis- 
ion ;  that  is.  the  difference  of  the  first  two  terms  is  to  the  first 
term,  as  the  difference  of  the  last  two  terms  is  to  the  third  term. 

Let  a  :  b  —  c  :  d. 

Then,  ad  =bc. 

Subtracting  both  members  from  ac, 

ac  —  ad  =  ac  —  be, 
or,  a{c  —  d)  =  c{a  —  b). 

Whence,  a  —  b:a  =  c  —  d:c. 

Similarly,  a  —  b  :  b  =  c  —  d  :  d. 

317.  /n  any  proportion  the  terms  are  in  proportion  by 
Composition  and  Division ;  that  is,  the  sum  of  the  first  two 
terms  is  to  their  difference,  as  the  sum  of  the  last  two  terms  is 
to  their  difference. 


(1) 

(2) 


Let 

a:b  —  c:d. 

Then  by  Art.  315, 

a  +  b      c-\-d 
a            c 

And  by  Art.  316, 

a— b      c—d 
a            c 

Dividing  (1)  by  (2), 

a-{-b      c-{-d 
a—b      c—d 

Whence,             a  +  b 

:  a—  b  —  c-\-d:  c 

d. 

318.  In  a  series  of  equal  ratios,  any  antecedent  is  to  its 
consequent,  as  the  sum  of  all  the  antecedents  is  to  the  sum  of 
all  the  consequents. 

Let  a:b  —  c:d=e:f. 

Then  by  Art.  309,  ad  =  be, 

and  a/=  be. 

Also,  ab  =  ba. 

Adding,  a(b-\-d -{•f)  =  b  (a+c  +  e). 

Whence  (Art.  312),       a:6  =  a  +  c  +  e:6  +  d  +/. 


RATIO   AND  PROPORTION.  271 

319.  In  any  number  of  proportions^  the  products  of  the 
corresponding  terms  are  in  proportion. 

Let  a:h  =  C'.d^ 

and  e  :  /=  g-.h. 

h      d  f     h 

Multiplying  these  equals, 

b     f     d      h'        bf     dh 
Whence,  ae  :  bf=  eg  :  dh. 

320.  In  any  proportion  ^  like  powers  or  like  roots  of  the 
terms  are  in  proportion. 

Let  a:b  =  c:d. 

rm,  a       c 

Therefore,  t-  =  t  • 

'  6*     d^ 

Whence,  a"  :  ft**  =  c"  :  d"". 

In  a  similar  manner  we  may  prove  that 

:^a'.yb  =  yc:  ^d. 

321.  In  any  proportion.,  if  the  first  two  terms  are  multiplied 
by  any  quantity.,  as  also  the  last  tivo,  the  resulting  quantities 
will  be  in  proportion. 

Let  a:b  =  c:d. 

Then, 

Therefore, 

Whence,  ma  :mb  =  nc:  nd. 


a 

—  = 

b 

c 
=  — . 

d 

ma 
mb 

_nc 
nd 

272  ALGEBRA. 

In  a  similar  manner  we  may  prove  that 

a     b  __c    d 
m'  m~  n'  n 

Note.  Either  m  or  n  may  be  unity ;  that  is,  either  couplet  may  be 
multiplied  or  divided  without  multiplying  or  dividing  the  otlier. 

322.  In  any  proportion,  if  the  first  and  third  terms  arh 
multiplied  by  any  quantity,  as  also  the  second  and  fourth 
terms,  the  resulting  quantities  will  be  in  proportion. 

Let  a  :  b  =  c  :  d. 

Then, 

Therefore, 

Whence,  ma  :  n6  =  mc  :  nd. 

In  a  similar  manner  we  may  prove  that 

a    b  _  c    d 
m'  n      m'  n 

Note.   Either  m  or  n  may  be  unity. 

323.  If  three  quantities  are  in  continued  proportion,  the 
first  is  to  the  third  as  the  square  of  the  first  is  to  the  square 
of  the  second. 


a  _ 

_  c 
~d 

ma 
nb 

_mc 

Let 

a:b  =  b:  c. 

Then, 

a      b 
b"  c 

Therefore, 

b      c~b      b 

Or, 

a^a^ 
c~b' 

Whence, 

a:c  =  a^'.W, 

RATIO  AND  PROPORTION.  273 

324.  If  four  quantities  are  in  continued  proportion^  the 
first  is  to  the  fourth  as  the  cube  of  the  first  is  to  the  cube  of 
the  second. 


Let  a:b  =  b:c  =  c:d. 

Then, 


^  — 5  —  ^. 
bed 


Therefore,  5x^x^  =  2x?x^. 

0      c     d      b      0      b 


Whence,  a:d  =  a^:  b^. 


Note.   The  ratio  a^ :  b^  is  called  the  duplicate  ratio,  and  the  ratio 
a* :  6^  the  triplicate  ratio,  of  a  :  6. 


PROBLEMS. 
325.   1.    Solve  the  equation, 

a;+l  :  X  —  1  =  a  -\-  b  :  a  —  b. 
By  Art.  317,  2a;:  2  =  2a:  26. 

Whence  by  Art.  321,     .t  :  1  =  a  :  6. 

Therefore,  x=  -■>   Ans. 

b 

2.    If   x:y  =  (x-{-zy:(y-{-zy,  prove   that  2;  is   a   mean 
proportional  between  x  and  y. 

From  the  given  proportion,  by  Art.  309, 

y{x-\-zy  =  x{y-\-zy. 

Or,  0!^y-\-2  xyz  -f-  yz"^  =  xy^-{-2  xyz  +  xz^. 

Or,  3?y  —  xy^  =  xz^  —  yz^. 

Dividing  by  a?  —  y,  ^  =  2;^. 

Therefore  2;  is  a  mean  proportional  between  x  and  y. 


274  ALGEBRA. 

3.  Find  the  first  term  of  the  proportion  whose  last  three 
terms  are  18,  6,  and  27. 

4.  Find  the  second  term  of  the  proportion  whose  first, 
third,  and  fourth  terms  are  4,  20,  and  55. 

5.  Find  a  fourth  proportional  to  |,  |^,  and  ^. 

6.  Find  a  third  proportional  to  f  and  |. 

7.  Find  a  mean  proportional  between  8  and  18. 

8.  Find  a  mean  proportional  between  14  and  42. 

9.  Find  a  mean  proportional  between  2^  and  -f^. 
Solve  the  following  equations  : 

10.  2ic-5:3aj  +  2  =  a;-l  :  7a;  +  l. 

11.  ar^-4:ar^-9  =  a^-5a;  +  6:aj2  +  4a;4-3. 


12.    x-i-Vl-x':x-VV^^  =  a+^/¥^^':a-VW' 


13.    \^''^--—        14. 
ix: 


x:y=    3:5.         --       (  x  +  y:  x  —  y=a  -{-b  :  a—b. 
4=15:1/.  ■     Xx'-^-f^a'b'ia'  +  b'). 


16.    Find  two  numbers  in  the  ratio  of  2|^  to  2,  such  that 
when  each  is  diminished  by  5,  they  shall  be  in  the  ratio  of 

H  to  1. 

16.  Divide  50  into  two  parts  such  that  the  greater  in- 
creased by  3  shall  be  to  the  less  diminished  by  3,  as  3  to  2. 

17.  Divide  12  into  two  parts  such  that  their  product  shall 
be  to  the  sum  of  their  squares  as  3  to  10. 

18.  Find  two  numbers  in  the  ratio  of  4  to  9,  such  that  12 
is  a  mean  proportional  between  them. 

19.  The  sum  of  two  numbers  is  to  their  difference  as  10 
to  3,  and  their  product  is  364.     What  are  the  numbers? 

20.  Ifa  —  6:&  —  c  =  6:c,  prove  that  6  is  a  mean  propor- 
tional  between  a  and  c. 


RATIO   AND  PROPORTION.  276 

21.  If  5a  +  46:  9a4-26  =  5  6  +  4c:96  4-2c,  prove  that 
6  is  a  mean  proportional  between  a  and  c. 

22.  If  (a  +  6+c  +  d)  (a-6-c  +  d)  =  (a-64-c-d) 
(a  +  6  —  c  —  d) ,  prove  that  a  :  h  =  c  :  d. 

23.  If  ax  —  by:cx—dy  =  ay  —  bz:cy  —  dz,  prove  that  y 
is  a  mean  proportional  between  x  and  z. 

24.  Find  two  numbers  such  that  if  3  is  added  to  each, 
they  will  be  in  the  ratio  of  4  to  3  ;  and  if  8  is  subtracted 
from  each,  they  will  be  in  the  ratio  of  9  to  4. 

25.  There  are  two  numbers  whose  product  is  96,  and  the 
difference  of  their  cubes  is  to  the  cube  of  their  difference  as 
19  to  1.     What  are  the  numbers? 

26.  Divide  $564  between  A,  B,  and  C,  so  that  A's  share 
may  be  to  B*s  in  the  ratio  of  5  to  9,  and  B's  share  to  C's  in 
the  ratio  of  7  to  10. 

27.  A  railway  passenger  observes  that  a  train  passes  him, 
moving  in  the  opposite  direction,  in  2  seconds  ;  whereas,  if 
it  had  been  moving  in  the  same  direction  with  him,  it  would 
have  passed  him  in  30  seconds.  Compare  the  rates  of  the 
two  trains. 

28.  Each  of  two  vessels  contains  a  mixture  of  wine  and 
water.  A  mixture,  consisting  of  equal  measures  from  the 
two  vessels,  contains  as  much  wine  as  water ;  and  another 
mixture,  consisting  of  four  measures  from  the  first  vessel 
and  one  from  the  second,  is  composed  of  wine  and  water  in 
the  ratio  of  2  to  3.  Find  the  ratio  of  wine  to  water  in  each 
vessel. 

29.  Divide  a  into  two  parts  such  that  the  first  increased 
by  b  shall  be  to  the  second  diminished  by  6,  as  a  +  3  Z)  is  to 
a -3b, 


276  ALGEBRA. 


XXIX.      VARIATION. 

326.  Ooe  quantity  is  said  to  vary  directly  as  another  when 
the  ratio  of  any  two  values  of  the  first  is  equal  to  the  ratio  of 
the  corresponding  values  of  the  second. 

Note.  It  is  customary  to  omit  the  word  *'  directly,"  and  say  simply 
that  one  quantity  varies  as  another. 

327.  Suppose,  for  example,  that  a  workman  receives  a 
fixed  sum  per  day. 

The  amount  which  he  receives  for  m  days  will  be  to  the 
amount  which  he  receives  for  n  days  as  m  is  to  n ;  that  is, 
the  ratio  of  any  two  amounts  received  is  equal  to  the  ratio  of 
the  corresponding  numbers  of  days  worked. 

Hence  the  amount  which  the  workman  receives  varies  as 
the  number  of  days  during  which  he  works. 

328.  One  quantity  is  said  to  vary  inversely  as  another 
when  the  first  varies  directly  as  the  reciprocal  of  the  second. 

Thus,  the  time  in  which  a  railway  train  will  traverse  a  fixed 
route  varies  inversely  as  the  speed ;  that  is,  if  the  speed  is 
doubled^  the  train  will  traverse  its  route  in  one-half  the  time. 

329.  One  quantity  is  said  to  vary  as  two  others  jointly 
when  it  varies  directly  as  their  product. 

Thus,  the  wages  of  a  workman  varies  jointly  as  the 
amount  which  he  receives  per  day,  and  the  number  of  days 
during  which  he  works. 

330.  One  quantity  is  said  to  vary  directly  as  a  second  and 
inversely  as  a  third,  when  it  varies  jointly  as  the  second  and 
the  reciprocal  of  the  third. 

Thus,  in  physics,  the  attraction  of  a  bod}^  varies  directly  as 
the  quantity  of  matter,  and  inversely  as  the  square  of  the 
distance. 


VARIATION.  ,      277 

331.  The  symbol  oc   is  used  to  express  variation ;  thus, 
a  ccb  is  read  "  a  varies  as  6." 

332.  If  xocy,  then  x  is  equal  to  y  multiplied  by  a  constant 
quantity. 

Let  a;'  and  ?/'  denote  a  fixed  pair  of  corresponding  values 
of  X  and  y,  and  x  and  y  any  other  pair. 
Then  from  the  definition  of  Art.  326, 

X      y  x' 

-  =  -,'    or  x  =  -y. 
x'      y'  y' 

x' 
Denoting  the  constant  ratio  —  by  m,  we  have 

x  =  my. 

333.  It  follows  from  Arts.  328,  329,  330,  and  332  that: 

1.  If  X  varies  inversely  as  y,  x=  — 

2.  If  X  varies  jointly  as  y  and  z,  x  =  myz. 

Z.    If  X  varies  directly  as  y  and  inversely  as  z,  x  =  — ^« 

z 

334.  Problems  in  variation  are  readily  solved  by  convert- 
ing the  variation  into  an  equation  by  aid  of  Arts.  332  or  333. 

EXAMPLES. 

335.  1.    If  a;  varies  inversely  as  y,  and  is  equal  to  9  when 
2/  =  8,  what  is  the  value  of  x  when  y—lS? 

If  X  varies  inversely  as  y,  we  have  by  Art.  333, 

m 
x  =  — 

y 

Putting  a;  =  9  and  y  =  8,  we  obtain 

9  =  -,    orm=72. 

8 

72 
Whence,  x  =  — 

y 

72 
Hence,  if  ?/  =  18,  we  have  x  =  —  =  4,  Ans. 

18 


278  ALGEBRA. 

2.  Given  that  the  area  of  a  triangle  varies  jointly  as  its 
base  and  altitude,  what  will  be  the  base  of  a  triangle  whose 
altitude  is  12,  equivalent  to  the  sum  of  two  triangles  whose 
bases  are  10  and  6,  and  altitudes  3  and  9,  respectively? 

Let  B,  H,  and  A  denote  the  base,  altitude,  and  area, 
respectively,  of  any  triangle,  and  B'  the  base  of  the  required 
triangle. 

Then  since  A  varies  jointly  as  B  and  H,  we  have 

A  =  mBH{Avt.  333). 

Therefore  the  area  of  the  first  triangle  is  m  X  10  X  3,  or 
30  m,  and  the  area  of  the  second  is  m  x  6  x  9,  or  54 ?n. 

Hence  the  area  of  the  required  triangle  is 

30m  +  54m,  or  84m. 
But  the  area  of  the  required  triangle  is  also  m  X  B'x  12. 
Therefore,  12mJB'=  84  m. 

Whence,  B'  =  7,  Ans, 

3.  If  yccx,  and  is  equal  to  36  when  a?  =  4,  what  is  its 
value  when  x—7? 

4.  If  yocz^^  and  is  equal  to  15  when  2=3,  what  is  the 
value  of  y  in  terms  of  z^? 

5.  If  X  varies  inversely  as  y^  and  is  equal  to  4  when  2/= 2, 
what  is  the  value  of  y  when  x  =  ^? 

6.  If  z  varies  jointly  as  x  and  2/,  and  is  equal  to  90  when 
x  =  S  and  2/  =  6,  what  is  the  value  of  z  when  x  =  2  and y  =7? 

7.  If  X  varies  directly  as  y  and  inversely  as  z,  and  is 
equal  to  4  when  y=2  and  2;  =  3,  what  is  the  value  of  x  when 
y  =  Sd  and  2;  =  15? 

8.  If2a;  —  3oc32/-f7,  and  x  =  3  when  y=l,  what  is  the 
value  of  x  when  ?/  =  —  1  ? 

9.  If  a^oc2/^  and  a;  =6  when  y  =  S,  what  is  the  value  of 
y  when  a;  =  2  ? 


VARIATION.  279 

10.  Two  quantities  vary  directly  and  inversely  as  x,  re- 
spectively. If  their  sum  is  equal  to  7  when  a;  =  2,  and  to 
—  13  when  a;=  —  3,  what  are  the  quantities? 

11 .  Given  that  the  volume  of  a  pyramid  varies  jointly  as 
its  base  and  altitude,  what  will  be  the  altitude  of  a  pyramid 
whose  base  is  12,  equivalent  to  the  sum  of  two  pyramids ' 
whose  bases  are  5  and  8,  and  altitudes  12  and  6,  respectively? 

12.  If  the  illumination  from  a  source  of  light  varies  in- 
versely as  the  square  of  the  distance,  how  much  farther  from 
a  candle  must  a  book,  which  is  now  3  inches  off,  be  removed 
so  as  to  receive  just  half  as  much  light? 

13.  Two  circular  plates  of  gold,  each  an  inch  thick,  the 
diameters  of  which  are  6  and  8  inches,  respectively,  are 
melted  and  formed  into  a  single  circular  plate  one  inch  thick. 
Find  its  diameter,  having  given  that  the  area  of  a  circle  varies 
as  the  square  of  its  diameter. 

14.  Three  spheres  of  lead  whose  diameters  are  3,4,  and  5 
inches,  respectively,  are  melted  and  formed  into  a  single 
sphere.  Find  its  diameter,  having  given  that  the  volume  of 
a  sphere  varies  as  the  cube  of  its  diameter. 

15.  If  5  men  in  6  weeks  earn  $57,  how  many  weeks  will 
it  take  4  men  to  earn  $  76  ;  it  being  given  that  the  amount 
earned  varies  jointly  as  the  number  of  men,  and  the  number 
of  weeks  during  which  they  work. 

16.  If  the  volume  of  a  cylinder  of  revolution  varies  jointly 
as  its  altitude  and  the  square  of  its  radius,  what  will  be  the 
radius  of  a  cylinder,  whose  altitude  is  18,  equivalent  to  the 
sum  of  two  cylinders  whose  altitudes  are  5  and  12,  and  radii 
6  and  9,  respectively? 

17.  Given  that  y  is  equal  to  the  sum  of  two  quantities,  of 
which  one  is  constant  and  the  other  varies  as  xy.  If  y  is 
equal  to  1  when  a;  =  —  2,  and  to  —  J  when  a?  =  2,  what  is  the 
expression  for  y  in  terms  of  x  ? 


280  ALGEBRA. 


XXX.    ARITHMETICAL    PROGRESSION. 

336.  An  Arithmetical  Progression  is  a  series  of  terms, 
each  of  which  is  derived  from  the  preceding  by  adding  a 
constant  quantity  called  the  common  difference. 

Thus,  1,  3,  5,  7,  9,  11,  ...  is  an  increasing  arithmetical 
progression,  in  which  the  common  difference  is  2. 

Again,  12,  9,  6,  3,  0,  —3,  ...  is  a  decreasing  arithmetical 
progression,  in  which  the  common  difference  is  —  3. 

337.  Given  the  first  term,  a,  the  common  difference,  d,  and 
the  number  of  terms,  n,  to  find  the  last  term,  I. 

The  progression  is 

a,  a-\-d,  a-{-2d,  a  +  3d,  ••• 

It  will  be  observed  that  the  coefficient  of  d  in  any  term  is 
one  less  than  the  number  of  the  term.  Hence,  in  the  nth,  or 
last  term,  the  coefficient  of  d  will  he  n—1.     That  is, 

l=a  +  {n-l)d.  (I.) 

338.  Given  the  first  term,  a,  the  last  term,  I,  and  the  num- 
ber of  terms,  n,  to  find  the  sum  of  the  series,  jS. 

S=a-j-(a-\-d)  +  {a-\-2d)-\ \-(l-d)-^L 

Writing  the  series  in  reverse  order, 

jS=1  +(l  -d)-^(l  -2d)-\-'--+(a-\-d.)  +  a. 
Adding  these  equations,  term  by  term, 

2S  =  (a+l)  +  (a+l)  +  (a+O  +  •••  +  (a+0  +  («+0 
=  n(a-}- 1). 

Therefore,  S  =  -(a  +  i).  (II.) 

339.  Substituting  in  (II.)  the  value  of  I  from  (I.) ,  we  have 

>S  =  |[2a-f-(r^-l)c^]. 


ARITHMETICAL  PROGRESSION.  281^ 


EXAMPLES. 

340.    1.  In  the  series  8,  5,  2,  —  1,  —4,  ...  to  27  terms, 
find  the  last  term  and  the  sum. 

In  this  case,  a  =  8,  d  =  — 3,  7i  =  — 27. 

Substituting  in  (I.)  and  (II.), 

Z  =  8 +  (27-1)  (-3)  =  8 -78  =  -    70. 

^  =  ?I(8-70)  =  27  X  (-31)  =-837. 

Note.   The  common  difference  may  be  found  by  subtracting  the 
first  term  from  the  second.     Thus,  in  the  series 

5  1  o  1  ;  1       5  11 

-, ,    —2,  •••,  we  have  a  = = 

3        6  6     3  6 

In  each  of  the  following,  find  the  last  term  and  the  sum  of 
the  series : 

2.  1,  6,   11,   ...  to  15  terms. 

3.  7,  3,  -1,  ...  to  20  terms. 

4.  -9,  -6,  -3,  ...  to  23  terms. 

5.  -5,  -10,  -15,   ...  to  29  terms. 

6.  -,    -,      ,  ...  to  35  terms. 

4     2     4 


7. 

?,    — ,  ...  to  19  terms. 
5     15 

8. 

|,    5,    5    ...  to  16  terms, 
3     4     6 

9. 

i     -^,  ...  to  22  terms. 

10. 

—  3,   — -,  ...  to  17  term 

11. 

2     1 
,      ,  ...  to  14  terms. 

5'    3' 


282  ALGEBRA. 

341.  If  any  three  of  the  five  elements  of  an  arithmetical 
progression  are  given,  the  other  two  may  be  found  by  sub- 
stituting the  given  values  in  the  fundamental  formulae  (I.) 
and  (II.)?  a^^  solving  the  resulting  equations. 

1.  Given  a  = ,n=20,S  = ;  find  d  and  Z. 

Substituting  the  given  values  in  (I.)  and  (II.) ,  we  have 

/  =  -2  +  19d.  (1) 

3  \     3       J  6  3  '  ^ 

From  (2),    Z  =  ^-l  =  l 
^  ^  3      6      2 

Substituting  in  (1), 

2  3  '         6 

Arts.  d=  -.  1  =  — 
6  2 

2.  Given  d  =  -3,  ?  =  -39,  aS  =  - 264;  find  a  and  w. 
Substituting  in  (I.)  and  (II.), 

_39  =  a  +  (^-l)(-3),  or  a  =  3n-42.  (1) 

-264  =  -(a-39),  or  an  -  39n  =  -  528.  (2) 

z 

Substituting  the  value  of  a  from  (1)  in  (2), 

3^2_  4271 -39n  =  - 528, 

or,  w2-27n  =  -176. 


Whence,  n  =  ^Z.±^^M=I1  =  ?7±^  =  16  or  11. 

2  2 

Substituting  in  (1), 

a  =48 -42,  or  33-42  =  6  or -9. 
Ans,  a=  6,  n  =  16  ;  or,  a  =  —  9,n=  11. 


ARITHMETICAL  PROGRESSION.  283 

Note.  The  interpretation  of  the  two  answers  is  as  follows : 

If  a  =  6,  and  n  =  16,  the  series  is 

0,  3,  0,  -  3,  -  6,  -  9,  - 12,  -  15,  -  18,  ^  21,  -  24, -  27,  -30, 
_33,  _36,  -39. 

If  a  =  —  9,  and  n  =  11,  the  series  is 

_  9,  _  12,  _  15,  -  18,  -  21,  -  24,  -  27,  -  30,  -  33,  -  36,  -  39. 
In  each  of  these  the  last  term  is  —  39,  and  the  sum  is  —  264. 

113 

3.    Givena  =  -,  d  = -,  S  =  —  -;  find  Z  and  w. 

o  1  ^  ^ 

Substituting  in  (I.)  and  (II.)  i 

3      ^  'V     12/  12 

-|  =  ^(^  +  '),      or    «  +  3;»  =  -9.  (2) 

Substituting  the  value  of  I  from  (1)  in  (2), 

5n-n^^_g     ^^   n2.-.9?i=36. 
4 
Solving  this  equation,  ?i  =  12  or  —  3. 

The  second  value  is  inapplicable,  for  the  number  of  terms 
in  a  progression  must  be  a  positive  integer. 

Substituting  the  value  n=  12  in  (1), 

5-12  7 


(1) 


12  12 

Ans.  Z  = -,  71  =  12. 

1  ^ 

Note.   A  negative  or  fractional  value  of  n  is  inapplicable,   and 
should  be  rejected  together  with  all  other  values  dependent  upon  it. 


EXAMPLES. 

4.  Given  c?  =  4,  ?  =  75,  ?i  =  19  ;  find  a  and  S. 

5.  Given  d  =  — l,n  =  15,  S  = ;  find  a  and  I. 


284  ALGEBRA.    "^ 

2 

6.  Given  a  = ,  w  =  18,  /  =  5  ;  find  d  and  8. 

o 

7.  Given  a  = ,  n=l^  JS  =  —7  ;  find  d  and  I. 

4 

8.  Givena  =  |,  Z  =  -^,  .^  =  -?|l;  finddandTi. 

9.  Given  Z  =  -31,  n=  13,  ^  =  - 169  ;  find  a  and  c«. 

10.  Given  d  =  -^,  S  =  -  328,  a  =  2  ;  find  Z  and  n. 

11.  Given  a  =  3,  Z  =  42| ,  cZ  =  2^  ;  find  n  and  S. 

12.  Given  d  =  —  4,  rj  =  17,  /S'  =  —  493  ;  find  a  and  /. 

13.  Given  Z  =  ^,  d  =  i,  >S  =  20  ;  find  a  and  n. 

14.  Given  Z  =  ^,  n  =  21,  /S^  —  ;  find  a  and  d. 

15.  Given  a  =  -l,  Z  =  --,  ;S  =  -  —  ;  find  d  and  w. 

o  o  3 

16.  Given  a  =  -  ?,  n  =  15,  /S  =  120  ;  find  d  and  l. 

4 

17.  Given  Z  =  -47,  d  =  -l,  ^^  =  -1118;  find  a  and  n. 

18.  Given  a  =  6,  d  =  -  -,  .S  =  -  —  ;  find  n  and  I. 

3  3 

From  (I.)  and  (II.)  general  formulae  for  the  solution  of 
cases  like  the  above  may  be  readily  derived. 

19.  Given  a,  d,  and  S ;  derive  the  formula  for  n. 
Substituting  the  value  of  I  from  (I.)  in  (II.), 

2/S  =  w[2a-h(n-l)cZ],  or  dn" ■^{'^a- d)n  =  2 S. 

This  is  a  quadratic  in  ti,  and  may  be  solved  by  the  method 
of  Art.  265. 


ARITHMETICAL  PROGRESSION.  285 

Multiplying  by  4d,  and  adding  (2  a  —  dy  to  both  members, 
4  dV  -{-4:d(2a-d)n  +  {2a-dy  =  8dS-^{2a^  dy. 

Extracting  the  square  root, 


2d7i  +  2a-d=±-V8dS  +  {2a-dy. 


Whence,  n 

20.  Given  a,   I,  and  n 

21.  Given  a,  ?i,  and  S 

22.  Given  d,  «,  and  S 

23.  Given  a,  c?,  and    / 

24.  Given  d,    /,  and  n 

25.  Given  Z,  7i,  and  >S' 

26.  Given  a,  d,  and  /S" 

27.  Given  a,    Z,  and  >iS' 

28.  Given  d,    Z,  and  >S' 


-  d-2a±-V8dS-\-{2a-dy^ 
2d 

derive  the  formula  for  d. 
derive  the  formulae  for  d  and  I. 
derive  the  formulae  for  a  and  I. 
derive  the  formulae  for  n  and  S. 
derive  the  formulae  for  a  and  S. 
derive  the  formulae  for  a  and  d. 
derive  the  formula  for  I. 
derive  the  formulae  for  d  and  n. 
derive  the  formulae  for  a  and  n. 


342.  To  insert  any  number  of  arithmetical  means  between 
two  given  terms. 

For  example,  let  it  be  required  to  insert  5  arithmetical 
means  between  3  and  —  5. 

This  signifies  that  we  are  to  find  an  arithmetical  progres- 
sion of  7  terms,  whose  first  term  is  3,  and  last  term  —  5. 

Substituting  a  =  3, 7  =  —  5,  and  ?i  =  7  in  (I.),  we  have 

_5  =  3  +  6d,  or  (i  =  -^- 


Hence  the  required  series  is 

o     5     1         . 
'33 


7         II 
3'    "T 


-5. 


28(3  ALGEBRA. 

343.  Let  X  denote  the  arithmetical  mean  between  a  and  6. 
Then,  by  the  nature  of  the  progression, 

jc  — a  =  6— ic,  or2ic=a-f-6. 

Whence,  aj  =  :^L±i. 

2 

That  is,   tlie  arithmetical  mean  between  two  quantities  is 
equal  to  one-half  their  sum. 

EXAMPLES. 

344.  1.  Insert  5  arithmetical  means  between  2  and  4. 

2.  Insert  7  arithmetical  means  between  3  and  —  1. 

3.  Insert  4  arithmetical  means  between  —  1  and  —  7. 

4.  Insert  6  arithmetical  means  between  —  8  and  —  4. 

1  13 

5.  Insert  8  arithmetical  means  between  -  and 

2  10 

Find  the  arithmetical  mean  between  : 

6.  24  and  -  14.  ^   ,  ,  ^      . 

7.  (a  +  &)'and-(a-6)2.  *    a-b^^    a-^b 

PROBLEMS. 

345.  1.  The  sixth  term  of  an  arithmetical  progression  is 

5  16 

-,  and  the  fifteenth  term  is  —     Find  the  first  term. 

6  3 

By   Art.   337,  the   sixth  term  is  a  +  6  rf,  and  the  fifteenth  term  is 
a+  14  c?;  hence, 

r«+  6rf=|.  (1) 

\a^Ud==^.  (2) 

Subtracting  (1)  from  (2) ,       9d  =  -,  oTd  =  -' 

2  ^ 

1  /»  jf 

Substituting  in  (2),  a  +  7  =  — ;  whence,  a  =  —  -,  Ans. 

3  3 

/ 


ARITHMETICAL  PROGRESSION.  287 

2.  Find  four  quantities  in  arithmetical  progression  such 
that  the  product  of  the  extremes  shall  be  45,  and  the  product 
of  the  means  77. 

Let  the  quantities  he  x —  Sy,  x  — y,  x  +  y,  and  x  +  Sy.    Then,  by  the 

conditions, 

rx2-9y2  =  45. 
\x^~    y^=ll. 

Solving  these  equations,  a:  =  ± 9  and  y  =  ±2. 

Therefore  the  quantities  are  3,  7,  11,  and  15;  or,  —  3,  —  7,  —  11,  and 
-15. 

Note.  In  problems  like  the  above  it  is  convenient  to  represent  the 
unknown  quantities  by  symmetrical  expressions.  Thus  if  five  quanti- 
ties had  been  required,  we  should  have  represented  thera  byx  — 2y, 
X  —  y,  x,  ar  +  y,  and  x  +  2  y. 

3.  Find  the  sum  of  the  odd  numbers  from  1  to  100. 

4.  The  seventh  term  of  an  ai-ithmetical  progression  is  27, 
and  the  thirteenth  term  is  •—  3.     Find  the  twenty -first  tenn. 

5.  Find  four  numbers  in  arithmetical  progression  such 
that  the  sum  of  the  first  and  third  shall  be  22,  and  the  sum 
of  the  second  and  fourth  36. 

6.  A  person  saves  $270  the  first  year,  $245  the  second, 
and  so  on.  In  how  many  years  will  a  person  who  saves 
every  year  $145  have  saved  as  much  as  he? 

7.  In  the  progression  m,  2m  — Sn,  3m  — 6 n,  ...  to  10 
terms,  find  the  last  term  and  the  sum  of  the  series. 

8.  The  seventh  term  of  an  arithmetical  progression  is 
5 a  +  4 6,  and  the  nineteenth  term  is  9a—  2b.  Find  the 
fifteenth  term. 

9.  Find  the  sum  of  the  even  numbers  beginning  with  2 
and  ending  with  500. 

10.  The  sum  of  the  squares  of  the  extremes  of  four  num- 
bers in  arithmetical  progression  is  200,  and  the  sum  of  the 
squares  of  the  means  is  136.     What  are  the  numbers? 


288  ALGEBRA. 

11.  The  seventh  term  of  an   arithmetical  progression  is 

1  3  9 

,  the  thirteenth  term  is  -,  and  the  last  term  is  —     Find 

2  2  2 

the  number  of  terms. 

12.  Find  five  quantities  in  arithmetical  progression  such 
that  the  sum  of  the  first,  third,  and  fourth  is  3,  and  the 
product  of  the  second  and  fifth  is  —  8. 

13.  Two  persons  start  together.  One  travels  10  leagues 
a  day ;  the  other  8  leagues  the  first  day,  which  he  augments 
daily  by  half  a  league.  After  how  many  days,  and  at  what 
distance  from  the  point  of  departure,  will  they  come  together? 

14.  A  body  falls  IQj^  ^^^^  ^^^^  ^^^*  second,  and  in  each 
succeeding  second  32|^  feet  more  than  in  the  next  preceding 
one.     How  far  will  it  fall  in  16  seconds? 

15.  Find  three  quantities  in  arithmetical  progression  such 
that  the  sum  of  the  squares  of  the  first  and  third  exceeds  the 
second  b}-  123,  and  the  second  exceeds  one-third  the  first 
by  6. 

16.  After  A  had  travelled  2|  hours  at  the  rate  of  4  miles 
an  hour,  B  set  out  to  overtake  him,  and  went  4|-  miles  the 
first  hour,  4|  the  second,  5  the  third,  and  so  on,  increasing 
his  speed  a  quarter  of  a  mile  every  hour.  In  how  many 
hours  would  he  overtake  A? 

17.  If  a  person  should  save  $100  a  year,  and  put  this  sum 
at  simple  interest  at  5  per  cent  at  the  end  of  each  year,  to 
how  much  would  his  property  amount  at  the  end  of  20 
years  ? 

18.  The  digits  of  a  number  of  three  figures  are  in  arith- 
metical progression ;  the  first  digit  exceeds  the  sum  of  the 
second  and  third  by  1  ;  and  if  594  be  subtracted  from  the 
number,  the  digits  will  be  inverted.     Find  the  number. 


GEOMETRICAL  PROGRESSION.  289 


XXXI.    GEOMETRICAL  PROGRESSION. 

346.  A  Geometrical  Progression  is  a  series  of  terms,  each 
of  which  is  derived  from  the  preceding  by  multiplying  by  a 
constant  quantit}'  called  the  ratio. 

Thus,  2,  6,  18,  54,  162,  ...  is  an  increasing  geometrical 
progression  in  which  the  ratio  is  3. 

Again,  9,  3,1,-,   -,  ...  is  a  decreasing  geometrical  pro- 

gression  in  which  the  ratio  is  — 

Negative  values  of  the  ratio  are  also  admissible ;  thus, 
—  3,  6,  —12,  24,  —48,  •••  is  a  geometrical  progression  in 
which  the  ratio  is  —  2. 

347.  Given  the  first  term,  a,  the  ratio,  r,  and  the  number 
of  terms,  n,  to  find  the  last  term,  I. 

The  progression  is  a,  ar,  ai^,  ar^,  ... 

It  will  be  observed  that  the  exponent  of  r  in  any  term  is 
one  less  than  the  number  of  the  term.  Hence,  in  the  ?ith  or 
last  term,  the  exponent  of  r  will  be  n  —  1 .     That  is, 

l  =  ar''~\  (I.) 

348.  Given  the  first  term,  a,  the  last  term,  I,  and  the  ratio, 
r,  to  find  the  sum  of  the  series,  S. 

/S  =   a-\-ar  +ar^-\ f-  ar""'^  +  ar""-^  -f-  ar""'^. 

Multiplying  each  term  by  r, 

rS  =  ar -\- a')'^ -\- ar^ -\ -f-  ar^~^  +  air^'^  -f  ar*". 

Subtracting  the  first  equation  from  the  second, 

rS-S  =  ar^^-a',    or,    S  =  ^''"  ~  ^. 

?•  —  1 
But  from  (I.),  Art.  347,  rl^ar"".     Hence, 

S  =  rLz^.  (II.) 

r  —  1 


290  ALGEBRA. 


EXAMPLES. 

349.    1.  In  the  series  3,  1,  -,  ...  to  7  terms,  find  the  last 
term  and  the  sum. 

In  this  case,  a  =  3,  r  =  -,  n  =  7.     Substituting  in  (1.)  and 
(11.),  ^ 


[sj      3'      24 


243 

ix— -3      —-3  21^6 

y_3      243         _729         _        729  ^  1093_ 

1  _  1  _  2  _  2  2^^  ' 

3  3  3 

Note.    The  ratio  may  be  found  by  dividing  the  second  term  by  the 
first. 

2.    In  the  series  —  2,  6,  —  18,  54,  ...  to  8  terms,  find  the 
last  term  and  the  sum. 

In  this  case,  a  =  —  2,  r  = =  —  3,  n  =  8.     Hence, 

—  z 

l  =  -2{-  Sy=  -  2  X  ( -  2187)  =  4374. 

^^  -3  X  4374 -(-  2)  ^  -13122  +  2^3^,gQ^       ^ 
—  3  —  1  —4 

In  each  of  the  following,  find  the  last  term  and  the  sum  of 
the  series : 

3.  1,  2,  4,   ...  to  9  terms. 

4.  3,  2,  -,  ...  to  7  terms. 

o 

6.    -2,  8,  -32,  ...  to  6  terms. 

6.  2,  -1,  i,  ...  to  10  terms. 

7.  ^^  Ti  Q»  •••  to  11  terms. 
>&     4     o 


GEOMETRICAL  PROGRESSION.  291 

8.  ?    -1,  -,  •••  to  8  terms. 

3  2 

9.  8,  4,  2,  •••  to  9  terms. 

10.  -, ,  — ,   •••  to  6  terms. 

4  4    12 

11.  3,  -6,   12,  ...  to  7  terms. 

12. , , ,  ••.  to  10  terms. 

3        3        6 

350.  If  any  three  of  the  five  elements  of  a  geometrical 
progression  are  given,  the  other  two  may  be  found  by  sub- 
stituting the  given  vahies  in  the  fundamental  formulae  (I.) 
and  (II.)?  and  solving  the  resulting  equations. 

But  in  certain  cases  the  operation  involves  the  solution  of 
an  equation  of  a  degree  higher  than  the  second ;  and  in 
others  the  unknown  quantity  appears  as  an  exponent,  the 
solution  of  which  form  of  equation  can  usually  only  be 
effected  by  aid  of  logarithms  (Art.  427). 

In  all  such  examples  in  the  present  chapter,  the  equations 
may  be  solved  by  inspection. 

1.    Given  a  =  —  2,  ?i  =  5,  /  =  -  32  ;  find  r  and  S. 
Substituting  the  given  values  in  (I.),  we  have 

—  32  =  —  2r* ;  whence,  r*  =  16,  or  r  =  ±  2. 
Substituting  in  (II.), 
Ifr=      2,  ^  =  ^r-^2)-^(-^  =-64  +  2  =  -62. 

Ifr  =  -2,  ^=(-^)^-^^>-(-^)=      64  +  2^_ 
-2-1  -3 

Ans.  r=2,  S  =  -62;  or,  r  =  -2,  >S'  =  -22. 

Note.   The  interpretation  of  the  two  answers  is  as  follows  : 
If  r  =     2,  the  series  is  —  2,  —  4,  —  8,  —  16,  -  32,  in  which  the  sum  is  —  (32. 
If  r=- 2,  the  series  is -2,     4,-8,     16,  -  32,  in  which  the  sum  is  -  22, 


292  ALGEBRA. 


2.    Given  a  =  3,  r  = ,  S  = ;  find  w  and  L 

3  729 


Substituting  in  (II.)? 

1640  3  Z4-9 


-1/-3 


729         _i_i  4 


Whence,  Z-f.9=^^;  or,  Z  = —- 

729  729 


Substituting  in  (I.)> 


=  3-^        ;  or, 


729         V      37  V      37  2187 

Whence,  by  inspection, 

n  —  1  =  7,  or  n  =  8. 

EXAMPLES. 

3.  Given  r=  2,  n=  10,  Z=  256  ;  find  a  and  S. 

no 

4.  Given  r  =  —  2,  n  =  6,  /S'=  —  ;  find  a  and  l. 

5.  Given  a  =  2,  n  =  7,  Z  =  1458  ;  find  r  and  S. 

6.  Given  a  =  1,  r  =  3,  Z  =  81 ;  find  n  and  S, 

7.  Given»'  =  i,n  =  8,^  =  ^^;  findaandL 

3  6561 

8.  Given  a  =  3,  ti  =  6,  I  — ;  find  r  and  S. 

1024 

1  127 

9.  Given  a=2,l  =— ,  >S'  =  ^^^  ;  find  n  and  r. 

32  32 

10.  Given  a  =  -,  r  =  -  3,  aS  =  -  91  ;  find  n  and  /. 

id 

11.  Given  Z=:--128,  r=2,  >S  =  -255;  find  a  and  ti. 


GEOMETRICAL  PROGRESSION.  293 

From  (I.)  and  (II.)  general  formulae  may  be  derived  for 
the  solution  of  cases  like  the  above. 


12.  Given  a,  r,  and  S 

13.  Given  a,    I,  and  S 

14.  Given  r,    I,  and  S 

15.  Given  ?•,  n,  and   I 

16.  Given  r,  ?i,  and  /S 

17.  Given  a,  w,  and  I 


derive  the  formula  for  I. 
derive  the  formula  for  r. 
derive  the  formula  for  a. 
derive  the  formulae  for  a  and  S. 
derive  the  formulae  for  a  and  I. 
derive  the  formulae  for  r  and  S. 


Note.  If  the  given  elements  are  n,  I,  and  S,  equations  for  a  and  r 
may  be  found,  but  there  are  no  definite  formuloe.  for  their  values.  The 
same  is  the  case  when  the  given  elements  are  a,  n,  and  aS^. 

The  general  formulae  for  n  involve  logarithms;  these  cases  are  dis- 
cussed in  Art.  427. 

351.  The  limit  (Art.  297)  to  which  the  sum  of  the  terms 
of  a  decreasing  geometrical  progression  approaches,  as  the 
number  of  terms  increases  indefinitely,  is  called  the  sum  of 
the  series  to  infinity. 

The  value  of  S  in  formula  (II.),  Art.  348,  may  be  written 

a  —  rl 


/S  = 


1 


In  a  decreasing  geometrical  progression,  the  greater  the 
number  of  terms  taken,  the  smaller  will  be  the  value  of  the 
last  term. 

Hence  as  the  number  of  terms  increases  indefinitely,  the 
term  rl  approaches  the  limit  0. 

\                                             ct  — —  rl  n 

Therefore  the  fraction approaches  the  limit 


1  —  r  1 —r 

That  is,  the  sum  of  a  decreasing  geometrical  progression 
to  infinity  is  given  by  the  formula 

a 


^=137-  ■      (I"-) 


294  ALGEBRA. 


EXAMPLES. 


1.    Find  the  sum  of  the  series  4,  —-,—,...  to  infiuit3^ 

Q 

In  this  case,  a  =  4,  r  =  —  -• 

3 

4  12 

Substituting  in  (III.)?  ^  = r  = — '  ^^'• 

1+-       ^ 
3 

Find  the  sum  of  the  following  to  infinity : 


2. 

2,  1,-,  ... 

'    '  2 

6. 

3    1    1 
4'2'3'- 

3. 

4,  -2,  1,  .., 

7. 

o         3       3 

^'     10'  100'  - 

4. 

-1    1    -^ 
''3'       9' 

... 

8. 

5        oO 

5. 

o        3 
-3,  -^,  - 

3 
25'-* 

9. 

'       ^'  ^^'  - 

352.    T(9  Jind  the  value  of  a  repeating  decimal. 

This  is  a  case  of  finding  the  sum  of  a  geometrical  pro- 
gression to  infinity,  and  may  be  solved  by  the  formula  of 
Art.  351. 

1.   Find  the  value  of  .85151  ... 

.85151  ...  =  .8+.051  +  .0005l4---- 

The  terms  after  the  first  constitute  a  decreasing  geometrical 
progression  in  which  a  =  .051,  and  r  =  .01. 

Substituting  in  (III.), 

^  ^  _:051_  ^^051  ^  _5]^  ^  J7 
l-.Ol       .99       990      330* 

Hence  the  value  of  the  given  decimal  is 

^      J^_281     . 
10  "^330"  330' 


GEOMETRICAL  PROGRESSION.  295 

EXAMPLES. 
Find  the  values  of  the  following : 

2.  .7272...  4.    .7333...  6.    .110303-.. 

3.  .407407...  5.    .52121...  7.    .215454... 

353.  To  insert  any  number  of  geometrical  means  between 
two  given  terms. 

Example.    Insert  4  geometrical  means  between  2  and 

This  signifies  that  we  are  to  find  a  geometrical  progression 

of  6  terms,  whose  first  term  is  2,  and  last  term 

243 

OA 

Substituting  a  =  2,  ri  =  6,  and  I  =  ^ —  in  (I.) ,  we  have 

64       „  ,       ,  ,32  ,  2 

— —  =  2r  \  whence,  r  = ■»    and  r  =  -. 

243  '243  3 

Hence  the  required  series  is 

«     4     8     16     32      64 

z,   -1    -?    — 1    — t    . 

3     9     27     81     243 

354.  Let  X  denote  the  geometrical  mean  between  a  and  b. 

Then,  by  the  nature  of  the  progression, 

X      b  2        7- 

-  =  -,    oYxr  —  ab. 

a     X 
Whence,  x  =  Va6. 

That  is,  the  geometrical  mean  bettveen  two  quantities  is  equal 
to  the  square  root  of  their  product. 

EXAMPLES. 

1  28 

355.  1.    Insert  6  geometrical  means  between  3  and  — -• 

729 

2.   Insert  5  geometrical  means  between  -  and  364|-. 


296  ALGEBRA. 

3.  Insert  6  geometrical  means  between  —  2  and  —  4374. 

3  3 

4.  Insert  7  geometrical  means  between  -  and 

2  512 

5.  Insert  5  geometrical  means  between  —  2  and  —  128. 

799 

6.  Insert  4  geometrical  means  between  3  and — 

^  1024 

Find  the  geometrical  mean  between  : 

7.  11|  and  2f 

8.  4a.-2+12a;2/  +  92/^  and4a.'2-12a;?/  +  9/. 

Q     a^  —  ah       -, 

9. and 


ah  +  6^         ah-  h^ 

PROBLEMS. 

356.  1.  Find  three  numbers  in  geometrical  progression, 
such  that  their  sum  shall  be  14,  and  the  sum  of  their  squares  84. 

Let  the  quantities  be  a,  ar,  and  ar^ ;  then,  by  the  conditions, 

f  a-^ar-\-  ar^      =  14.  (1) 

'^  \  a2  +  a2r2  +  aV  =  84.  (2) 

Dividing  (2)  by  (1),  a-~ar-\-  ar^  =  6.  (3) 

Subtracting  (3)  from  (1),  2ar  =  S,  or  r  =  --  (4) 

a 

Substituting  in  (1),  a  +  4  +  —  rz:  14. 

a 

Or,  a2  _  10  a  =  -  16. 

Solving  this  equation,  a  =  8  or  2. 

Substituting  in  (4),  r  =  -  or  -  =  ^  or  2. 

8       2      2 

Therefore,  the  numbers  are  2,  4,  and  8. 

2.  The  fifth  term  of  a  geometrical  progression  is  48,  and 
the  eighth  term  is  —  384.     Find  the  first  term. 

3.  The  sum  of  the  first  and  second  of  four  quantities  in 
geometrical  progression  is  15,  and  the  sum  of  the  third  and 
fourth  is  60.     What  are  the  quantities? 


GEOMETRICAL  PROGRESSION.  297 

4.  Find  three  quantities  in  geometrical  progression,  such 
that  the  sum  of  the  first  and  second  is  20,  and  the  third 
exceeds  the  second  by  30. 

5.  The  fourth  term  of  a  geometrical  progression  Is  —  108, 
and  the  eighth  term  is  —  8748.     Find  the  first  term. 

6.  A  person  who  saved  ever}'  year  half  as  much  again  as 
he  saved  the  previous  year,  had  in  seven  years  saved  $2059. 
How  much  did  he  save  the  first  year? 

7.  The  elastic  power  of  a  ball,  which  falls  from  a  height 
of  a  hundred  feet,  causes  it  to  rise  to  0.9375  of  the  height 
from  which  it  fell,  and  to  continue  in  this  way  diminishing 
the  height  to  which  it  will  rise,  in  geometrical  progression, 
until  it  comes  to  rest.     How  far  will  it  have  moved? 

8.  The  sum  of  four  quantities  in  geometrical  progression 
is  30,  and  the  quotient  of  the  fourth  quantity  divided  by  the 

4 
sum  of  the  second  and  third  is  —     Find  the  quantities. 

9.  The  third  term  of  a  geometrical  progression  is  — ,  and 

the  sixth  term  is Find  the  eighth  term. 

512 

10.  Divide  the  number  39  into  three  parts  in  geometrical 
progression y  such  that  the  thii'd  part  shall  exceed  the  first 
by  24. 

11.  The  product  of  three  numbers  in  geometrical  progres- 
sion is  64,  and  the  sum  of  the  squares  of  the  first  and  third 
is  68.     What  are  the  numbers? 

12.  The  product  of  three  quantities  in  geometrical  pro- 
gression is  8,  and  the  sum  of  their  cubes  is  73.  What  are 
the  quantities? 


298  ALGEBRA. 


XXXII.    HARMONICAL    PROGRESSION. 

357.  Quantities  are  said  to  be  in  Harmonical  Progression 
when  their  reciprocals  form  an  arithmetical  progression. 

Thus,  1,  -,  -,  -,  -,  ...  are  in   harmonical   proojression, 
''3579  ^    "^ 

because  their  reciprocals  1,  3,  5,  7,  9,  ...  form  an  arithmetical 

progression. 

358.  Any  problem  in  harmonical  progression,  which  is 
susceptible  of  solution,  may  be  solved  by  taking  the  recipro- 
cals of  the  terms  and  applying  the  formulae  of  the  arithmet- 
ical progression. 

There  will  be  found,  however,  no  general  formula  for  the 
sum  of  the  terms  of  a  harmonical  series. 

359.  Let  X  denote  the  harmonical  mean  between  a  and  b. 

Then,  by  the  nature  of  the  progression,  -  is  the  arithmet- 

11  ^ 

ical  mean  between  -  and  — 
a  b 

Whence,  1  =.  ^_^  (Art.  343)  =  ^+^. 

X         2      ^  ^        2ab 

Therefore,  x  = 

a  +  b 

360.  If  any  three  consecutive  terms  of  a  harmonical  series 
are  taken,  the  first  is  to  the  third  as  the  first  minus  the  second 
is  to  the  second  minus  the  third. 

Let  the  terms  be  a,  6,  and  c. 

Then  since  -,  -,  and  -  are  in  arithmetical  progression, 
a    b  c 

c      b      b      a 

b  —  c      a  —  b 
or,  -— -  =  — — . 

be  ab 


HARMONICAL   PROGRESSION.  299 


Multiplying  both  members  by ,  we  have 

a  _a  —  b 
c      b  —  c 


EXAMPLES. 

2    2 
361.    1.    In  the  series  2,  -,  -,  »..  to  36  terms,  find   the 

last  term. 

Taking  the  reciprocals  of  the  terms,  we  have  the  arith- 
metical progression 

1     3     5 

2'    2'    2'  '" 

In  this  case  a  =  -,  cZ=  1,  n  =  36. 
Substituting  in  (I.)»  -^.rt.  337,  we  have 

;  =  |  +  (36-l)xl  =  ^. 

2 
Taking  the  reciprocal  of  this,  we  obtain  -^  as  the  last  term 

of  the  given  harmonical  series. 

2.    Insert  5  harmonical  means  between  2  and  —  3. 

Taking  the  reciprocals  of  the  terms,  we  have  to  insert  5 

arithmetical  means  between  -  and 

2  3 

Substituting  a  =  -,  Z  = ,  and  n  =  7,  in  (I.) ,  Art.  337, 

2  o 

we  have 

—  _  =  -4-6a;  or,  a  = 

3      2  36 

Then  the  arithmetical  series  is 

1    15    ?    Jl     _i_    _-I     _i. 
2'    36'    9'    12'        18'        36'        3* 

Therefore  the  required  harmonical  series  is 

2,    ^A    ?,    12,    -18,    -55,    -3. 
13     2         '  7 


300  ALGEBRA. 

Find  the  last  terms  of  the  following : 

3.  -,,^ — ,   -,  ...  to  11  terms. 
4     11     6 

4.  -,    --,    --,...  to  17  terms. 

5-    -5   -,   —  ?  •..  to  23  terms. 
2     3      8 

6. , , ,  ...  to  26  terms. 

3         2  7 

7.    —-,——-, ,  ...  to  31  terms. 

7         23         16 

2  3 

8.  Insert  7  harmonical  means  between  -  and  — 

5  10 

9.  Insert  4  harmonical  means  between  —  2  and  —  8. 

10.  Insert  6  harmonical  means  between  3  and  —  1. 
Find  the  harmonical  mean  between  : 

11.  3  and  -  5.  12.   ?^+^  and  ^!^. 

a  —  b  a  +  b 

13.  Find  the  last  term  of  the  harmonical  series 

a,  6,  ...  to  n  terms. 

14.  If  m  harmonical  means  are  inserted  between  a  and  by 
what  is  the  second  mean  ? 

15.  The  fourth  and  ninth  terms  of  a  harmonical  progres- 

3              1 
sion  are and ,  respectively.      What  is   the   seventh 

term  ? 

16.  Prove  that  the  geometrical  mean  between  two  quanti- 
ties is  a  mean  proportional  between  their  arithmetical  and 
harmonical  means. 


THE  BINOMIAL   THEOREM.  301 

XXXIII.    THE    BINOMIAL    THEOREM. 

POSITIVE    INTEGRAL    EXPONENT. 

362.  The  Binomial  Theorem  is  a  formula  by  means  of 
which  any  power  of  a  binomial  may  be  expanded  into  a  series. 

Examples  of  its  application  have  been  given  in  Art.  196. 

Proof  of  the  Theorem  for  a  Positive  Integral 
Exponent. 

363.  If  we  assume  the  laws  of  Art.  196  to  hold  for  the 
expansion  of  (a  +  x)""-,  where  n  is  any  positive  integer  : 

The  exponent  of  a  in  the  first  term  is  n,  and  decreases  by 
1  in  each  succeeding  term. 

The  exponent  of  x  in  the  second  term  is  1 ,  and  increases 
by  1  in  each  succeeding  term. 

The  coefficient  of  the  first  term  is  1  ;  of  the  second  term, 
n  ;  multiplying  n,  the  coefficient  of  the  second  term,  by  n  —  1, 
the  exponent  of  a  in  that  term,  and  dividing  the  result  by  the 

exponent  of  x  increased  by  1 ,  or  2,  we  have     ^^^  ~ — -  as  the 

coefficient  of  the  third  term  ;  and  so  on. 

Thus,  (a-\-xy  =  a^-{-  na"-^a;  _^n(7t  -  1)  ^„_2^ 
»  V     I     /  ^  ^      1.2 

n(n-l)(n-2)^._3^^  (1) 

1.2.3  ^  ^ 

This  result  is  called  the  Binomial  Theorem. 
Multiplying  both  members  by  a  +  «,  we  have 

^    ^    ^  ^  1.2 

,  w(n  — l)(n  — 2)    __2   Q  , 
+    ^      1.2.3 -''"^+- 

+  a"x  +  na"-'ar=  +  'ii^Jziia'-^a;'  +  ... 

X  *  if 


302  ALGEBRA. 

Collecting  the  terms  which  contain  like  powers  of  a  and  x, 
(a  +  g;)"+^  =  a"+^  +  {n  +  l)a*'x  +  r^(^~ ^)  +n   a''-^^^ 

rn(n-l)(yi-2)      n(n-l)~|       2  , 
L  1.2.3  1.2     J 

=  a"+i  +  (n  +  l)a»aj  +  nf  ^^^^  +  l1  a'*"  V 


^'— ^  +  1  |a--=^ar^  +  ... 


1-2 

=  a'^+i  +  (n  H-  l)a"a'  +  nf^^^lla'^-ia^ 


+ 


n(n—  1)  fn  +  1' 
1.2      L     ^ 


=  a'^+i  +  (n  +  l)a"cc  +  (^  +  l)^^n-ia^ 

1 .  z 

(7i  +  l)n(n-l)^,_,^ 
1.2.3 

It  will  be  observed  that  this  result  is  in  accordance  with 
the  laws  of  Art.  196. 

Hence,  if  the  laws  of  Art.  196  hold  for  any  power  of  a  +  a; 
whose  exponent  is  a  positive  integer,  they  also  hold  for  an 
exponent  greater  by  1. 

But  in  Art.  196,  the  laws  were  shown  to  hold  for  (a  -\-xy, 
and  hence  they  also  hold  for  {a-\-x)^;  and  since  they  hold 
for  (a  +  xy,  they  also  hold  for  (a-\-xy;  and  so  on. 

Therefore  the  laws  hold  when  the  exponent  is  any  positive 
integer,  and  equation  (1)  is  proved  for  any  positive  integral 
value  of  n. 

Note  1.  The  above  method  of  proof  is  known  as  the  Method  of 
Induction. 

Note  2.  In  place  of  the  denominators  1-2,  1  •  2  •  3,  etc.,  it  is  cus- 
tomary to  write  [2,  [3,  etc.  The  symbol  \n,  read  "factorial  n,"  signifies 
the  product  of  the  natural  numbers  from  1  to  n  inclusive. 


THE  BINOMIAL  THEOREM.  303 

I 

364.  Putting  a=  1  in  equation  (1),  Art.  363,  we  obtain 

EXAMPLES. 

365.  Note.  The  Notes  on  page  164  apply  with  equal  force  to  the 
examples  in  the  present  chapter.  If  the  second  term  of  the  binomial 
is  negative,  it  is  convenient  to  enclose  it,  negative  sign  and  all,  in  a 
parenthesis,  before  applying  the  laws  of  Art,  196.  In  reducing  after- 
wards, care  must  be  taken  to  apply  the  principles  of  Art.  192. 

1.    Expand  (m"^  —  V^)*- 

(m~^  -  -^ny  =  [(m"^)  +  ( -  w^)]*. 

The  exponent  of  (m~^)  in  the  first  term  is  5,  and  decreases 
by  1  in  each  succeeding  term. 

The  exponent  of  (— n^)  in  the  second  term  is  1,  and 
increases  by  1  in  each  succeeding  term. 

The  coefficient  of  the  first  term  is  1 ;  of  the  second  term, 
5  ;  multiplying  5,  the  coefficient  of  the  second  term,  by  4, 

the  exponent  of  {m~^)  in  that  term,  and  dividing  the  result 

by  the  exponent  of  {  —  n^)  increased  by  1,  or  2,  we  have  10 
as  the  coefficient  of  the  third  term  ;  and  so  on.     Hence, 

■i-10(m~^y(-7i^y+5{m~^){-n^y-{-(-n^y 
—  m~^—  5m~^n-  +  10m~^7i  —  lOm'^n^ 
-{-6m~^n^—n^,  Ans. 
Expand  the  following : 

2.  (ct  +  c^^)^  4.  f^-^'. 

3.    (m~^-ny.  5.    {x"'  +  2y^y. 


304  ALGEBRA. 

6.    (a«  +  3V^)'.  12.    (^  +  -1- 


7.    ,^^-V^Y.  13. 


-M 


8.    ^i^  +  il-Y-  ^^-    («^*  +  32/"*)^ 


.w^^y 


?r 


9.      m^-^    .  15. 


a  Vft      2  Va;\^ 


2/  V2a;^        a^h^ 

10.  (aU-*-a-^6^)''.  16.    (3  a-'\/^  -  ^"^  ^«)*- 

11.  (Va^-3>y.  17.  (yll+^yl'ij' 

Note.  A  trinomial  may  be  raised  to  any  power  by  the  Binomial 
Theorem  if  two  of  its  terms  are  enclosed  in  a  parenthesis  and  regarded 
as  a  single  term.     (Compare  Art.  195.) 

Expand  the  following : 

18.  (i-x-ay^y.  20.    (l  +  2aj-a^)*. 

19.  (a^  +  aj_2)4.  21.    (1-x  +  a^y, 

366.  To  find  the  rth  or  general  term  in  the  expansion  of 
(a-\-xy. 

The  following  laws  will  be  observed  to  hold  for  any  term 
in  the  expansion  of  {a-\-  x)'\  in  equation  (1),  Art.  363  : 

1.  The  exponent  of  x  is  less  b}^  1  than  the  number  of  the 
term. 

2.  The  exponent  of  a  is  7i  minus  the  exponent  of  x. 

3.  The  last  factor  of  the  numerator  is  greater  by  1  than 
the  exponent  of  a. 

4.  The  last  factor  of  the  denominator  is  the  same  as  the 
exponent  of  x. 

Therefore,  in  the  rth  term,  the  exponent  of  x  will  be  r  — 1. 
The  exponent  of  a,  will  be  n  —  (r  —  1) ,  or  w  — r  +  !• 
The  last  factor  of  the  numerator  will  be  7i  —  r  +  2. 
The  last  factor  of  the  denominator  will  be  r  —  1. 


THE   BINOMIAL   THEOREM.  306 

Hence,  the  rth  term 

1.2.3...(r-l) 

EXAMPLES. 
3G7.    1.  Find  the  eighth  term  of  (3a^  -  b'^y^. 

In  this  case,  r  =  8,  and  ti  =  11  ;  hence  the  eighth  term 
^11.10.9.8.7.6.5        K.,^,y 

=  330(81  a-){-b-'')  =  -  26730a26-%  Ans. 

Note.   If  the  second  term  of  the  binomial  is  negative,  it  should  be 
enclosed,  sign  and  all,  in  a  parenthesis,  before  applying  the  formula. 

Find  the 

2.  Seventh  term  of  (a  +  a^)"; 

3.  Sixth  term  of  (l  +  m)^«. 

4.  Eighth  term  of  (c  -  dy\ 
6.    Fifth  term  of  {l-ay\ 

6.  Seventh  term  of  (  -  +  -  )  • 

\b      a) 

7.  Fifth  term  of  {x--yjxy^. 

8.  Sixth  term  of  fa" "^ ab]. 


9.    Eighth  term  of  («-^+ 22/^)  1^ 

10.  Fourth  term  of  (a'  -3a;-^)". 

/  2  \^ 

11.  Ninth  term  of  (  V^i  +  -: —  )  • 


306  ^  ALGEBRA. 

XXXIV.    THE    THEOREM    OF    UNDETER- 
MINED   COEFFICIENTS. 

368.  A  Series  is  a  succession  of  terms  so  related  that  each 
may  be  derived  from  one  or  more  of  the  others  in  accord- 
ance with  some  fixed  law. 

The  simpler  forms  of  series  have  already  been  exhibited  in 
the  progressions. 

369.  A  Finite  Series  is  one  having  a  finite  number  of 
terms. 

An  Infinite  Series  is  one  the  number  of  whose  terms  is 
unlimited. 

The  progressions  in  general  are  examples  of  finite  series  ; 
but  in  Art.  351  we  considered  infinite  geometrical  series. 

370.  Infinite  series  may  be  developed  by  the  process  of 
Division,  when  the  divisor  is  not  exactly  contained  in  the 
dividend. 

Let  it  be  required,  for  example,  to  divide  1  by  1  —  x. 

1-x)  1  (l4-a;  +  aj2+ar^-f  ... 

1-x 


X 

X- 

-x' 

x' 

x'-a^ 

Therefore,      =  1-f  a;  +  a^  +  ic^H 

1  —  x 

Infinite  series  may  also  be  obtained  by  the  process  of  Evo- 
lution (see  Examples  20  to  23,  page  169),  and  by  other 
methods,  one  of  the  most  important  of  which  will  be  consid- 
ered in  Art.  376. 


UNDETERMINED  COEFFICIENTS.  307 

371.  A  series  is  said  to  be  convergent  either  when  the  sum 
of  the  first  n  terms  approaches  a  certain  fixed  quantity  as  a 
limit  (Art.  297),  when  n  is  indefinitely  increased,  or  when 
the  sura  of  all  the  terms  is  equal  to  a  finite  quantity. 

A  series  is  said  to  be  divergent  when  the  sum  of  the  first  n 
terms  can  be  made  to  numerically  exceed  any  assigned  quan- 
tity, however  great,  by  taking  n  sufficiently  great. 

372.  Consider,  for  example,  the  infinite  series 

I.  Suppose  X  =  iCi,  where  Xi  is  positive  and  <  1. 
The  sum  of  the  first  n  terms  is  now 

1  -f-  a.'i  +  a:,2  +  0^1^  H-  . . .  +  xr'  =  V^^        (^'•*-  ^^) 

1  —  Xi 

As  n  increases  indefinitely,  a^i*  decreases  indefinitely,  and 
approaches  the  limit  0. 

1    /-  n  J^ 

Therefore  the  fraction ^  approaches  the  limit 

I  —Xi  1  —  JCl 

That  is,  the  sum  of  the  first  n  terms  approaches  a  certain 
fixed  quantity  as  a  limit,  when  n  is  indefinitely  increased. 
Hence  the  series  is  convergent  when  x  is  positive  and  <  1. 

II.  Suppose  a;  =1. 

In  this  case,  each  term  of  the  series  is  equal  to  1,  and  the 
sum  of  the  first  n  terms  is  equal  to  n ;  and  this  sum  can  be 
made  to  numerically  exceed  any  assigned  quantity  however 
great,  by  taking  n  sufficiently  great. 

Hence  the  series  is  divergent  when  x=l. 

III.  Suppose  ic  >  1 . 

In  this  case,  each  term  of  the  series  after  the  first  is  >1, 
and  the  sum  of  the  first  w  terms  is  >  n  ;  and  this  sum  can  be 
made  to  numericall}'  exceed  any  assigned  quantity  however 
great,  by  taking  n  sufficiently  great. 

Hence  the  series  is  divergent  when  a;  is  >  1. 


308  ALGEBRA. 

373.  If  an  infinite  series  is  convergent,  the  greater  the 
number  of  terms  taken,  the  more  nearly  does  their  sum  ap- 
proach to  the  value  of  the  expression  which  produced  the 
series  ;  but  if  it  is  diveYgent,  the  sum  diverges  more  and  more 
from  the  value  of  the  expression. 

Consider,  for  example,  the  equation  (Art.  370), 

— J_=l  ^x  +  X^  +  Otf-i-". 
1  —X 

Putting  a;=.l,  in  which  case  the  series  is  convergent 
(Art.  372),  the  equation  becomes 

y=l  +  .l  +  .01  +  .001  +  - 

In  this  case,  however  great  the  number  of  terms  taken,  the 

sum  can  never  be  made  exactly  equal  to  — ,  but  it  approaches 

y 
this  value  as  a  limit.     (See  Art.  352.) 

Again,  putting  x=  10,  in  which  case  the  series  is  diver- 
gent, the  equation  becomes 

-  i  =  1  +  10  -f  100  -}-  1000  -f  .- 
y  „^ 

In  this  case  it  is  evident  that  the  sum  of  the  terms  diverges 

more  and  more  from  the  value 

9 

It  follows  from  the  above  that  an  infinite  series  cannot  be 
regarded  as  representing  the  value  of  the  expression  which 
produced  it,  unless  it  is  convergent. 

374.  The  infinite  series 

a -\- bx -}- cx^  -\-  da^  -\ 

is  convergent  when  a;  =  0 ;  for  the  sum  of  all  the  terms  is 
equal  to  a  when  x  =  0. 

THE  THEOREM  OF  UNDETERMINED   COEFFICIENTS. 

375.  An  important  method  for  expanding  expressions 
into  series  is  based  on  the  following  theorem,  known  as  the 
Theorem  of  Undetermined  Coefficients. 


UNDETERMINED  COEFFICIENTS.  309 

376.  If   the    series   A-\- Bx-[- Coi? -\- Dd^  -\ is    always 

equal  to  the  series  A'  +  B'x  +  C'x^  -f  D'x^  -\ ,  when  x  has 

any  value  which  makes  both  series  convergent,  the  coefficients 
of  like  powers  of  x  in  the  two  series  will  be  equal;  that  is, 
A  =  A',  B  =  B\  C=C\  etc. 

For  since  the  equation 

A-it  Bx  +  Cx"  ^  Da?  -i- '-  =  A^ -{-  B'x  +  Ox"  +  D'^ff'  +  '" 

is  satisfied  when  x  has  any  vaUie  which  makes  both  series 
convergent,  and  since  both  members  are  convergent  when 
x  =  0  (Art.  374),  it  follows  that  the  equation  is  satisfied 
when  ic=  0. 

Putting  a;  =  0,  we  have  A  =  A'. 

Subtracting  A  from  the  first  member  of  the  equation,  and 
its  equal  A'  from  the  second  member,  we  obtain 

jBa;  +  Ca^  4-  ^ar^  +  •••  =  B'x  +  C'a^  +  i)'ar^  +••• 

Dividing  through  by  x, 

B-\-Cx-^Dx^-\-'-'  =  B'-\-  C'x  +  D'x'+'" 

This  equation  also  is  satisfied  when  x  has  any  value  which 
makes  both  members  convergent ;  and  i^utting  a;  =  0,  we  have 

B  =  B', 

In  like  manner  we  may  prove  C  =  C',  D  =  D',  etc. 

Note.  The  reason  for  limiting  the  theorem  to  values  of  x  which 
make  both  series  convergent,  is  that  a  convergent  series  evidently 
cannot  be  equal  to  a  divergent  series;  and  two  divergent  series  cannot 
be  equal,  because  two  expressions  neither  of  which  is  finite  cannot  be 
said  to  be  equal. 

377.  Since  a  finite  series  is  always  convergent,  it  follows 
from  the  preceding  article  that  if  two  finite  series 

A  +  Bx-\-Cx^-}- ...  4-  -ffic"  and  A'-\-  B'x-^C'x^-{ \-K'x\ 

are  equal  for  every  value  of  x,  the  coeflEiciepts  of  like  powers 
of  X  in  the  two  series  are  equal, 


310 


ALGEBRA. 


APPLICATION  TO  THE  EXPANSION  OF  FRACTIONS 
INTO  SERIES. 


378.    1.  Expand 


Sx" 


X' 


l-2x-hSx' 


in  ascending  powers  of  x. 


We  have  seen  in  Art.  370  that  a  fraction  of  the  above  form 
can  be  expanded  into  a  series  b^^  dividing  the  numerator  by 
the  denominator ;  we  therefore  know  that  the  proposed  ex- 
pansion is  possible. 


Assume  then 


2-3i»2 


^A  +  Bx+Cx'-^-Dx^-^-  Ex'  + 


(1) 


l-2a;-f-3a;2 

where  A,  B,  C,  D^  E,  ...,  are  quantities  independent  of  x. 

Clearing  of  fractions,  and  collecting  the  terms  in  the  second 
member  involving  like  powers  of  x,  we  have 


2-3x'-a^  =  A-\-     B 
-2A 


x+    C 

x^-h    D 

x'^-    E 

-2B 

-20 

-2D 

+  3^ 

+  35 

+  30 

a;^+...   (2) 


The  second  member  of  (1)  must  express  the  value  of  the 
fraction  for  every  value  of  x  which  makes  the  series  con- 
vergent (Art.  373). 

Hence  equation  (2)  is  satisfied  when  x  has  any  value 
which  makes  both  members  convergent,  and  by  the  Theorem 
of  Undetermined  Coefficients,  the  coefficients  of  like  powers 
of  X  in  the  two  series  must  be  equal ;  that  is, 

A=     2. 

B-2A=     0;  whence,  5=  2 J.  =4. 

0-2J5+3^=-3;  whence,  0=  25-3^- 3  =  -  1. 

D- 2  0  +  35=-!;  whence,  D=20-3  5- 1  =- 15. 

jS;-2Z)+30=     0;  whence, -E^=  2 i>-3  0        =-27;  etc. 


UNDETERMmED   COEFFICIENTS.  311 

Substituting  these  values  in  (1),  we  have 

^-^x'-x^   ^2  +  4.x-:x^-lbx^-21x' ,  Ans. 

\-2x+2>a? 

The  result  may  be  verified  by  division. 

Note.  A  vertical  line,  called  a  bar,  is  often  used  instead  of  a  paren- 
thesis; thus, 

+     5  I  X  is  equivalent  to  (^  —  2  A)  x. 

-2a\ 

If  the  numerator  and  denominator  contain  only  even  pow- 
ers of  a;,  the  expansion  will  involve  only  even  powers  of  x ; 
in  this  case  the  operation  may  be  abridged  by  assuming  a 
series  containing  only  the  even  powers  of  x. 

Thus,  if  the  fraction  were  — — — — -,  we  should  assume 

l-^x'+bx^ 

it  equal  io  A-^Ba?-^  Cx*  -\- Dx^ -\- Ex^  +  • - 

In  like  manner,  if  the  numerator  contains  only  odd  powers 
of  ic,  and  the  denominator  only  even  powers,  we  should 
assume  a  series  containing  only  the  odd  powers  of  x. 

If  every  tenn  of  the  numerator  contains  x,  we  may  assume 
a  series  commencing  with  the  lowest  power  of  x  in  the  nu- 
merator. 

EXAMPLES. 

Expand  each  of  the  following  to  five  terms,  in  ascending 


powers  of  x 

2. 

l-x 

1-^x 

3. 

2  +  5a: 

1-Sx 

A 

S-4.a^ 

l+5»2 

n 

2x 

X  —  9t? 


l+X  +  X^ 


10. 


M      X  —  o  Xi   ~~'  Xj  ^  ^ 


1  -  «2 


12. 


9         I  ~  ^^  2^3 

3-2ar^  *    \-^2x-Z;ff  '    ^-Zx-^x^ 


2-2>x^-\^ 

\-\-2x-hx? 

:^^-2:^ 

2-x-x^ 

3^_a;-2»» 

3_a;2-}-a53 

l-3ar^ 

312 


ALGEBRA. 


If  the  lowest  power  of  x  in  the  denominator  is  higher  than 
the  lowest  power  in  the  numerator,  we  may  determine  by 
actual  division  what  power  of  x  will  occur  in  the  first  term  of 
the  expansion  ;  we  should  then  assume  the  fraction  equal  to 
a  series  commencing  with  this  power  of  a;,  the  exponents  of 
X  in  the  succeeding  terms  increasing  by  unity  as  before. 

1 


14.    Expand 


^x'-o? 


in  ascending  powers  of  x. 


X 

Dividing  1  by  3  ar,  the  quotient  is  '- —  ;  we  then  assume 


1 


Clearing  of  fractions, 
l  =  3vl  +  3J5 


=  Ax~''  +  Bx-^^  C+Dx  +  Ex^-\- 


(1) 


a;  4-30 
-    B 


ar^  +  3Z> 

-     C 


-    D 


Equating  the  coefficients  of  like  powers  of  », 
3^=1 
^B-A=0 
SC-B=0 
3D-G=0 
SE~-D=0:  etc. 


Whence, 

A 

4- 

=!■- 

1 

=  27' 

--k 

Substituting 

in  (1), 

we  have 

1 

=f- 

.-    1 

9        27 

^i 

+  243  + 

3a^- 

a^ 

243 


Ans. 


etc. 


Expand  each  of  the  following  to  five  terms,  in  ascending 
powers  of  a;: 

2  ,«     l-2a^-ir» 


16. 
16. 


3ar'-4a^ 
x-2af  +  3i^' 


17. 
18. 


x^  -\-a^  —  x^ 
3-2a;4-a;^ 


2a^ 


2a;« 


UNDETERMINED  COEFFICIENTS. 


313 


APPLICATION    TO    THE    EXPANSION    OF    RADICALS 
INTO    SERIES. 


379.    1.    Expand  Vl  —x  in  ascending  powers  of  x. 

We  have  seen  in  Art.  204  that  the  square  root  of  an  imper- 
fect square  can  be  expanded  into  a  series  by  the  process  of 
Evolution  ;  we  therefore  know  that  the  proposed  expansion 
is  possible.     Assume  then 

(1) 


VT^^  =  A  -^  Bx  -\-  Ca^  -{•  Dx^  -\-  Ex*  -^  '" 
Squaring  both  members,  we  have  by  Art.  194, 


l-x  =  A' 


+  2AB 


x^       ^ 

x" 

X'  +        C 

+  ^AC 

+  2  AD 

+  2AE 

+  2BC 

+  2BD 

X*  ■■{-'" 


Equating  the  coefficients  of  like  powers  of  a;, 
A^=     1;  whence,  ^=1. 


whence,  B=  — 
whence,  C=  — 


J_ 
2A 

2A 


1  n  J5C  1 

whence,  i>= = 

A  16 


whence,  E= 


C^±2BD 
2A 


128 


2AB^-\ 

B^-j-2AC=     0 

2AD-j-2BC=     0 

C'-h2AE-\-2BD=    0 
etc. 

Substituting  these  values  in  (1),  we  have 

rz ^         X        X^         X^         5x*  . 

~         2      8       16      128  ' 

The  result  may  be  verified  by  the  method  of  Art.  204. 

Note.   The  equation  A^=\  gives  A=  ±\;    and  taking  the  negative 

value  of  A,  we  should  find  B=~f    C=  ->   D  =  — ,    etc. 

2  8  16 

Thus  another  answer  to  the  example  is 


814  .  ALGEBRA. 


EXAMPLES. 


Expand  each  of  the  following  to  five  terms,  in  ascending 
powers  of  x : 


2.    Vl+2a;.  4.    Vl-2ajH-3a;^         6.    -Vl-x. 


3.    Vl  -Sx.  5.    Vl  +  x-af.  7.    s/l+a;-har^. 

APPLICATION    TO  THE    DECOMPOSITION   OF  RATIONAL 
FRACTIONS. 

380.  If  the  denominator  of  a  fraction  can  be  resolved  into 
factors,  each  of  the  first  degree  in  ic,  and  the  numerator  is  of 
a  lower  degree  than  the  denominator,  the  Theorem  of  Unde- 
termined Coeflflcients  enables  us  to  express  the  given  fraction 
as  the  sum  of  two  or  more  j)ortial  fractions^  whose  denomi- 
nators are  factors  of  the  given  denominator,  and  whose 
numerators  are  independent  of  x. 

Case  I. 

381.  When  no  two  factors  of  the  denominator  are  equal. 

19ic  -4-  1 

1.    Separate  — into  partial  fractions. 

^  (3a;-l)(5a;  +  2)  ^ 

Assume ' = ,  (1) 

(3a;- 1)  (5a; +  2)      3a;-l      5a;4-2  ^^ 

where  A  and  B  are  quantities  independent  of  x. 

Clearing  of  fractions,  we  have 

19a; -f  1  =  ^  (5a;  +  2)  + -S  (3a;  -  1) 

=  {5A  +  SB)x  +  2A-B.  (2) 

The  second  member  of  equation  (1)  must  express  the 
value  of  the  given  fraction  for  every  value  of  x. 

Hence  equation  (2)  is  satisfied  by  every  value  of  x,  and 
by  Art.  377  the  coefficients  of  like  powers  of  x  in  the  two 
members  are  equal. 


UNDETERMINED  COEFFICIENTS.  315 

That  is,  5^  +  3jB  =  19, 

and  2A-    B=    1. 

Solving  these  equations,  we  obtain  A  =  2  and  B  =  S, 
Substituting  in  (1),  we  have 

19-  +  1         _      2        ,       3         ^„^_ 


(3.r-l)(5a;4-2)      Sx-1      5x  +  2 
The  result  may  be  verified  by  adding  the  partial  fractions. 

X  -\-  4: 

2.    Separate  -~ — —  into  partial  fractions. 

ji  X  —  io    —  2/ 

The  factors  of  2  a;  —  a.*^  —  af^  are  a;,  1  —  a;,  and  2  -f-  a;  (Art. 
283).     Assume  then 


2x  —  a^  —  a?      X      1  —  x      2-^x 
Clearing  of  fractions,  we  have 

X -\-  4:  =  A{l-x)  {2  -^ x)  +  Bx  (2  -I- a;)  4-  Cx(\-x) . 

This  equation,  being  satisfied  b}^  every  value  of  a;,  is  satis- 
fied when  a;  =  0. 

Putting  a;  =  0,  we  have       4  =  2^,  or  A  =  2. 

Again,  the  equation  is  satisfied  when  a;  =  1. 

Putting  a:  =  1,  we  have       5  =  3  .B,  or  B  =  -. 

o 

The  equation  is  also  satisfied  when  a;  =  —  2. 
Putting  a;  =  —  2,  we  have  2  =  —  6  (7,  or  C= 

o 

Substituting  in  (1),  we  obtain  , 

5  _1 

a;  +  4  2         3  3 

! =  -  _i I 

2a;  — a:^  — a:^      x      1  —  x      2-fa; 

^a"*"3(l-a;)~3(2+a;)' 

Note.  The  student  should  compare  the  above  method  of  finding  A 
and  B  with  that  used  in  Example  1. 


816  .      ALGEBRA. 

EXAMPLES. 
Separate  the  following  into  partial  fractions : 

g  18  a; +10  g     2a;^-  17a;-24 

7  ixo/  —  .t-a^LO  -  ^  2  iC^  —  20 


3. 

Ux 

-25 

Aa^ 

-25 

A 

4x 

+  15 

Sx^ 

+  bx 

H 

x"- 

-45 

13  0^+10 

6i 

r^_13ar-5a; 

aa.'-Ua2 

a.-2 

-3aic-4a2 

7.T  +  9 

8.   ''"-^'^      .       11. 


(a^-4)(a^-l) 
4ic-14 


2a^-18a;  9  +  9aj-4aj2  4a;2-20a;  +  23 

Case  II. 

382.     Wlien  all  the  factors  of  the  deriominator  are  equal. 

x^ 11a;  +  26 

Example.    Separate  - —  ^ —  into  partial  fractions. 

(x  —  3) 

If  we  attempt  to  solve  the  example   by   the  method   of 
Case  I.,  we  should  assume 

a^-lla;+26_     A  B  C 


{x-Zy  ic-3      aj-3      x-'6 

That  is,  ^^-1^^  +  ^^  =  A±A±G, 

{x-2>y  x-z 

But  this  is  evidently  impossible,  for  the  given  fraction 
cannot  be  reduced  to  an  equivalent  fraction  having  a;  —  3  for 
a  denominator,  and  a  numerator  independent  of  x. 

Let  us  now  substitute  in  the  given  fraction  2/  +  3  in  place 
of  a; ;  we  then  have 

(y  +  3)^-ll(y  +  3)+26^y^-5y  +  2^1      5    ^    2 
yi  f  y      y^      f 

Replacing  2/  by  a;  —  3,  the  result  takes  the  form 

I 5  2 

x-2,      {x-^y      {x-^y 


UNDETERMINED  COEFFICIENTS.  317 

This  shows  that  the  given  fraction  can  be  expressed  as 
the  sum  of  three  partial  fractions,  whose  numerators  are  in- 
dependent of  X,  and  whose  denominators  are  the  powers  of 
x  —  3  beginning  with  the  first  and  ending  with  the  third. 

A  similar  result  will  hold  in  any  example  under  Case  II.  ; 
the  number  of  partial  fractions  being  equal  to  the  number  of 
equal  factors  in  the  denominator  of  the  given  fraction. 

EXAMPLES. 
383.    1.  Separate — — ■  into  partial  fractions. 

In  accordance  with  the  principle  stated  in  Art.  382,  we 
assume  the  given  fraction   equal  to  the  sum  of  two  partial 
fractions,  whose  denominators  are  the  powers  of  3  a;  +  5  be- 
ginning with  the  first  and  ending  with  the  second;  that  is, 
6a;  +  5    ^      A  B 

{Sx-j-5y~3x-\-5       {3x-j-5y 
Clearing  of  fractions,  we  have 

6a;  +  5  =  ^(3a;H-5)  -\- B 
=  SAx-\-5A+B. 
Equating  the  coeflficients  of  like  powers  of  x, 
SA=6, 
and  5A-^B  =  6.  - 

Solving  these  equations,  we  have  A  =  2  and  B  =  —  5. 

Whence,        6fl;  +  5        _J 5 ^^^ 

(3a;+5)2      3^.4.5      (Saj  +  S)^' 

Separate  the  following  into  partial  fractions  : 

2  2a;-13  ^     3x^-4.  '      g     x(5x-4) 
a.-2-f-10a^+25*          *    {x-\-iy  '     {5x-2y' 

3  ^  5     18a;^+12a;-3         ,-     x(x-\-2y 

{x-2y  '     {3x-\-2y   '       '    (x-{.iy' 

g     2a^-10a.-^+17a;-10         g     4a^-18a^ 
{x-iy  '  '     {2x-3y' 


318  ALOEBRA. 


Case  III. 


384.    When  some  of  the  factors  of  the  denominator  are 
equal. 

1.    Separate — — ^into  partial  fractions. 

X\,X  ~\~  J.  ) 

The  method  in  Case  III.  is  a  combination  of  the  methods 
of  Cases  I.  and  II.  ;  we  assume 


x{x-\-iY    X    x  +  i    {x  +  \y    {x  +  iy 

Clearing  of  fractions, 
^x  -\-2=  A{x  +  ly  +  Bx{x  +  iy  +  Cx{x  -\-l)  +  Dx 
=  ( J.  +  B)x^  +  (3^  +  25  +  C)x^ 
+  {^A-\-B+C+D)x  +  A, 
Equating  the  coefficients  of  like  powers  of  «, 

A-\-B  =  0, 

3^  +  2J5+(7=0, 

3u4  +  5+(7  +  Z)  =  3, 

and  A  =  2. 

Solving  these  equations,  we  have 

^=  2,  JB  =  -2,  (7=-2,  andZ)=l. 
Substituting  in  (1), 

3a;  +  2        2  2  2,1  . 


x{x-^\y    X    x  +  i    {x  +  \y    {x-\-\y 

Note.   It  is  impracticable  to  give  an  illustrative  example  for  every 

possible  case ;  but  the  student  should  find  no  difficulty  in  assuming  the 

proper  partial  fractions  if  attention  is  given  to  the  following  general 

rule : 

jr 

A  fraction   of  the  form should  be    put 

equalto  (.  +  a)(.  +  .) ...  (.  +  m)'... 


X  •\-a       X  ■\-h  ar+jw       (ar+  w)^  (^  +  wj)' 

Single  factors  like  x  •\-  a  and  x  -{-h  having  single  partial  fractions 

corresponding,  arranged    as   in   Case   I. ;    and   repeated   factors   like 

(^x-\-m)r  having  r  partial  fractions  corresponding,  arranged  as  in  Case  II. 


UNDETERMINED  COEFFICIENTS.  319 

EXAMPLES. 

Separate  the  following  into  partial  fractions : 

S-^x-x"  g     3a^-llar^  +  13a;-4 

*     x{x-\-2y'  '       x{x-l){x-2y 

3       3a;-l  g     15  -  7a;  +  3a^- 3a^ 

x'ix-j-iy  '  a;*  +  5cc3 

(2a;-3)(2a:2_7^^g)*        *      sc»(^x-hiy 

385.   If  the  degree  of  the  numerator  is  equal  to,  or  greater 

than,  that  of  the  denominator,  the  preceding  methods  are 

inapplicable. 

x^  —  Sa^  —  1 

Thus,  let  it  be  required  to  separate into  partial 

ar  —  X 
fractions. 

If  we  proceed  as  in  Case  I.,  we  should  assume 
x^-Sx^-l  ^A        B 


3^  —  X  X         X—1 

Clearing  of  fractions  and  uniting  terms, 

:x?-^x'-lz={A-\-B)x-A. 
Equating  the  coefficients  of   q(?^  we  have  1  =  0,  a  result 
which  shows  that  the  method  of  Case  I.  is  inapplicable. 

But  by  actual  division,  we  obtain 

—   =  a;-2+— -.  (1) 

XT  —  X  XT  —  X 

2x  —  1 

We  can  now  separate  — into  partial  fractions  by 

xr  —  x 

the  method  of  Case  I. ;  the  result  is 
1         3 
X     x  —  1 
Substituting  in  (1),  we  have 

—  =  x  —  2-] •>   Ans. 

or  —  X  X     X  —  I 


820 


ALGEBRA. 


EXAMPLES. 

Separate  the  following  into  entire  quantities  and  partial 
fractions : 


1. 


8a^-36a;^-2 
(2rc-5)(2a;  +  l) 


3     5a^+5af  —  2x^-^S 
x^'  +  af 


5. 


4     3ar^-2a^  +  22a^  +  9; 
{x'-iy 

2x^-2a^-7x^-i-2a^  +  x-l 
x^  —  oc^ 


APPLICATION  TO  THE  REVERSION  OF  SERIES. 

386.    Note.    To  revert  a  given  series  y  —  a  +  bx'^  +  cx^  +  ...  is  to 
express  x  in  terms  of  y. 

Example.    Revert  the  series 

y  =  2x  +  x^-2a^-Sx* -{-'•» 

Assume  x  —  Ay-\-  By^  -\-  Cy^  +  Dy^  -\ (1) 

Substituting  in  this  the  given  value  of  2/,  we  have 
a;  =  ^  (2a; -f- a;2  -  2a;3  _  3  aj4  4- ...) 

+  5  (4a;2  +  a-^  +  4.^3  -  8  a;^  +  ...) 
4-0(8a.'3  +  12a;^4-...) 
+  i)(16a;*+. ..)+••• 

x^-\ 


That  is,  a;  =  2  ^ar  4-    A 
+  45 

a?-2A 
+  45 

+  8(7 

x^-    3  A 
-    IB 
+  12  C 
+  16i) 

Equating  the  coefficients  of  like  powe 

2^=1 

rs  of  a;, 

-2^  +  4 
~3^-75  +  12C 

^  +  45  = 
5  +  80  = 
^+16i)  = 

=  0 
=  05 

=  0; 

etc. 

u:n^determined  coefficients.  321 


Solving  these  equations, 

^  =  2'  ^  =  -8'^  =  ^'^  =  - 

■S'  - 

Substituting  in  (1),  we  have 

r      8^  ^16^        128^^ 

•  ♦,   Ans. 

If  the  eyen  powers  of  x  are  wanting  in  the  given  series, 
the  operation  may  be  abridged  by  assuming  x  equal  to  a 
series  containing  only  the  odd  powers  of  y. 

Thus,  to  revert   the  series  y  =  x  —  '3i?-\-x'  —  i^-\ ,   we 

should  assume 

x=^Ay^Bf■\-C]t^-Dy'^-•" 

If  the  odd  powers  of  x  are  wanting  in  the  given  series, 
the  reversion  of  the  series  cannot  be  effected  by  the  method 
previously  given.  But  by  substituting  another  letter,  say  t^ 
for  x^,  we  ma}'  revert  the  series  and  express  t  in  terms  of  y ; 
and  by  taking  the  square  root  of  the  result,  x  itself  may  be 
expressed  in  terms  of  y. 

If  the  first  teim  of  the  given  series  is  independent  of  a;,  it 
is  impossible,  by  the  method  previously  given,  to  express  x 
definitely  in  terms  of  y  ;  but  it  is  possible  to  express  it  in  the 
form  of  a  series  in  which  y  is  the  only  unknown  quantity. 

Let  it  be  required,  for  example,  to  revert  the  series 

y=2^-2x-^y?-2^-Zx''^"' 

The  series  may  be  written 

1/ -  2  =  2a;  4- «2  -  2a^  -  Sx*  +  •.. 

We  then  assume 

a;  =  ^(2/-2)  +  5(2/-2)=^-hC(2/-2)3+i)(2/-2)^+... 

Proceeding  as  in  Ex."  1,  we  find 

a,=  1(2,-2)  -i(y-2)^  +  A  {y-%Y-  ^  (2/-2)<+  ... 


322  ALGEBRA. 

EXAMPLES. 
387.    Revert  each  of  the  following  to  four  terms 
1.    y  =  x-\-3(^  +  x^-\-x^-\ 

*   ^      2       4         6         8 

4.   2/  =  l4-a;  +  ^  +  -  +  -+  — 

I?      |3      li 

6.   y  =  x  —  af-\-x^  —  x'^-\-"' 

6.  2/  =  ^-^  +  ^-^'  +  .- 
^      2      3       4       5 

7.  2/  =  3a;4-5a^+7a^+lla;^-t-  — 
^  3^5       7 


THE  BINOMIAL   THEOREM.  323 

XXXV.    THE    BINOMIAL    THEOREM. 

FRACTIONAL    AND    NEGATIVE    EXPONENTS. 

388.  It  was  proved  in  Art.  364  that  when  n  is  a  positive 
integer, 

(1+  xy  =  l-^nx-\--^-- — ^a^H — ^^ -^ ^af +  ...      (1) 

11  '_ 

Proof  of  the  Theorem  for  any  Exponent. 

389.  I.  When  the  exponent  is  a  positive  /inaction. 

Let  the  exponent  be  ^,  p  and  q  being  positive  integers. 


Then,  (l  +  a;)?"  =  V(l +a;)^  (Art.  218) 


=  ^l+pa;+-,by  (1). 
It  is  evident  that  a  process  may  be  found,  analogous  to 

those  of  Arts.  203  and  208,  for  expanding  -^1  -\-px-\ in 

ascending  powers  of  x ;  and  the  first  term  of  the  result  will 
evidently  be  1.     Assume  then. 


Vl-fpa;+---  =  l+3fa;4-^ar^  +  --  (2) 

Raising  both  members  to  the  ^th  power,  we  have 
1  -^px-{- '"  =[1  -{.{Mx  -h  Nx^  -\-  "•)2' 

=  l-\-q{Mx+Nx--i-  ...)  +  -,  by  (1). 

This  equation  being  satisfied  by  every  value  of  x  which 
makes  both  members  convergent,  by  the  Theorem  of  Unde- 
termined Coefficients  (Art.  376)  the  coefficients  of  x  in  the 
two  series  are  equal. 

That  is,  p  =  qM,  or  J/  =  -• 

Substituting  this  value  in  (2) ,  we  have 

{l-{-xy=l+^x+...  (3) 


324  ALGEBRA. 

II.     When'the  exponent  is  a  negative  quantity. 

Let  the  exponent  be  —  s,  s  being  a  positive  quantity. 

Then,  (1  +  x)-'  =        ^         (Art.  221) 

(1  -j-X)" 

^  by  (1)  or  (3). 


l+sx-\- 
Whence  by  actual  division,  we  obtain 

{l+x)-'=l-sx+"'  (4) 

From  (1),  (3),  and  (4),  we  observe  that  whether  n  is 
positive  or  negative,  integral  or  fractional,  the  form  of  the 
expansion  is 

(1  +  «)"  =  1  4-  waj  +  ^o.-^  -j-  JBa^  +  ...  (5) 

X 

Writing  -  in  place  of  x,  we  obtain 


(-!)■ 


a         a-         a^ 


Multiplying  both  members  by  a*",  we  have 

(a  +  a;)"  =  a''  +  7ia''^^x  4-  Aa^'-^x^  +Ba''-^x'  +  ...       (6) 

This  result  is  in  accordance  with  the  second,  third,  and 
fourth  laws  of  Art.  196 ;  hence  these  three  laws  hold  for  any 
value  of  the  exponent. 

390.  We  will  now  prove  the  Jifth  law  of  Art.  196  for  any 
value  of  the  exponent. 

Let  P  and  Q  denote  the  coefficients  of  ic''  and  x'"+^  in  the 
second  member  of  (5)  ;  then  (5)  and  (6)  may  be  written 

(1  4-  aj)"  =  1  +  7ix  +  ...  +  iV  +  Qx^-^'  +  ... ,  (7) 

and 

(a  4-  xy  =  a"+  na'^-^x  -\ \- Pa'"'' x' -^  Qa''-'-^ x'+'^ -] (8) 

In  (8)  put  a=l  -\-y,  and  x  =  z',  then, 
(1  4-2/  +  ^)^  =  (l  4-2/)*^  4- '"  4-P(l  4-2/)'*-'-=?'-4-  ••'  (») 


THE  BINOMIAL  THEOREM.  325 

Again,  in  (7)  put  x  —  z-\-y;  then, 
(1  +z  +  yY  =  1  +  -  +  P(z  +  yy  +  Q{z-{-yy-^'-^  •>' 

Expanding  the  powers  of  z  +  yhj  aid  of  (8),  we  have 
{1  -hz  -{-yy  =  1  -h  '"  +  P[z^  -hrz^~'y  +  '-'] 

+  Q  [2'-+'+ (r  +  l)2^'-y +•••]+•••    (10) 

The  first  members  of  (9)  and  (10)  being  identical,  their 
second  members  are  equal  for  every  value  of  z  which  makes 
both  series  convergent ;  and  by  the  Theorem  of  Undeter- 
mined Coefficients,  the  coefficients  of  z"  in  the  two  series  are 
equal ;  that  is, 

^(1  +  y)""''  =  P+  Q(r  +  1)2/  +  terms  in  /,  f,  etc. 

Expanding  the  first  member  by  aid  of  (7),  this  becomes 

P[H-C/» -r)  2/ +•••]  =  ^+Q(r  4-1)2/  + - 

This  equation  being  satisfied  by  every  value  of  y  which 
makes  both  members  convergent,  the  coefficients  of  y  in  the 
two  series  are  equal. 

Therefore,  F{n-r)=  Q(r  +  1),  or  Q  =  ^-=^. 

That  is,  the  coefficient  Q  is  equal  to  the  coefficient  of  the 
preceding  term  in  (8),  multiplied  by  the  exponent  of  a  in 
that  term,  and  divided  by  the  exponent  of  x  increased  by  1. 

Thus  the  fifth  law  of  Art.  196  is  proved  to  hold  for  any 
value  of  the  exponent. 

391.  By  aid  of  the  law  proved  in  Art.  390,  the  coefficients 
of  the  terms  after  the  second  in  the  second  member  of  (8) , 
Art.  390,  may  be  readily  found  as  in  (1),  Art.  363. 

Thus,   {a  +  xy  =  a^-^  na'^-'  x  +  ^'^'^  ~  ^^  ct'^-^ar^ 

L? 

and  the  Binomial  Theorem  is  proved  in  its  most  general  form. 


326  ALGEBRA. 

If  n  is  a  positive  integer,  the  number  of  terms  in  the  series 
is  w  +  1  ;  for  all  coefficients  after  the  (?i+l)st  contain  the 
factor  n  —  n,  or  0.      (Compare  Art.  196.) 

But  if  n  is  fractional  or  negative,  the  expansion  never 
terminates,  since  no  one  of  the  quantities  /i—  1,  n  — 2,  ..., 
can  become  equal  to  zero.  The  development  in  this  case 
furnishes  an  infinite  series,  which  however  expresses  the 
value  of  (a  ■j-x)''  only  for  such  values  of  a  and  x  as  make 
the  series  convergent.     (Compare  Art.  373.) 

EXAMPLES. 

392.  In  expanding  expressions  by  the  Binomial  Theorem 
when  the  exponent  is  fractional  or  negative,  it  is  convenient 
to  obtain  the  exponents  and  coefficients  of  the  terms  by  aid 
of  the  laws  of  Art.  196,  which  have  been  proved  to  hold  uni- 
versally. 

If  the  second  term  is  negative,  it  should  be  enclosed,  sign 
and  all,  in  a  parenthesis,  as  in  Arts.  365  and  367,  before 
applying  the  laws. 

1.    Expand  (a  +  x)^  to  four  terms. 

2 
The  exponent  of  a  in  the  first  term  is  -,  and  decreases  by 

o 

1  in  each  succeeding  term. 

The  exponent  of  x  in  the  second  term  is  1 ,  and  increases 
by  1  in  each  succeeding  term. 

The  coefficient  of  the  first  term  is  1  ;  of  the  second  term, 

2  2                                                                            1 
- ;  multiplying  -,  the  coefficient  of  the  second  term,  by , 

3  3  o 

2 
the  exponent  of  a  in  that  term,  and  dividing  the  product,  —  -, 

J 
by  the  exponent  of  x  increased  by  1,  or  2,  we  have  —  -  as 

the  coefficient  of  the  third  term  ;  and  so  on.     Hence, 

(a-tx)^  =  a^-i--a~^x  —  -ar^i^-\ a~^  x^ .  Ans. 

^  ^  3  9  81 


THE  BINOMIAL  THEOREM.  327 

2.    Expand  (1  —  2a;~2)-2  to  ^^^  terms. 

-  4.1-^.  (-2a;"2)3_|.  5.1-6. (_2x-^)4-.. . 
1 


Mi-i 


3.    Expand  j^=z  to  four  terms. 
yia'^-^3x^ 

-v/a-^  +  3x^      (a-^  +  3a;^)^ 

-M(ai)-¥(3.^)3+... 

=  a^  — a^a;- 4-2a^a;  — —  a'^^x^H ,   ^?is. 

o 


Expand  each  of  the  following  to  five  terms : 

\- 

6.    {1-x)-^.      11.    {x--^-Sy)^.       16 


4.    {a-\-x)K  9.    ?— — •  14.    (a;<  +  4a6)^. 

(a  —  aj)3 


1 


7.    V^^=^.  12.    (a-2a^)-i        17.    (4a2+a;-^)i 

393.  The  formula  for  the  rth  term  of  (a  +  ^y  (Art.  366) 
holds  for  fractional  and  negative  values  of  n,  since  it  was 
derived  from  an  expansion  which  has  been  proved  to  hold 
universally. 


328  ALGEBRA. 

EXAMPLES. 
1.   Find  the  seventh  term  of  (a  —  3a;~^)"^. 

(a  -Sx~^y^  =  [a  -\-(-3x~^)yK 

In  this  case  r  =  7,  and  n  = ;  hence  the  seventh  term 

o 


10        13        16 


1.2.3.4.5.6 

Find  the 

2.  Eighth  term  of  (a  4-  x)K 

3.  Twelfth  term  of  (1  +  m)-*. 

4.  Fifth  term  of  (l-a'yK 

5.  Seventh  term  of  (a  —  x)K 

6.  Sixth  term  of  (a*  +  b^yK 

7.  Seventh  term  of  (ic^i  —  ^"^)^. 

8.  Sixth  term  of 

9.  Eleventh  term  of  (a^  +  2x)K 
10.  Ninth  term  of 


3      _19  a    „ 

—  a  5  (_3a;-2)6 


1 1 .    Sixth  term  of  (a^  +  3  x'^)  ~i 


12.   Eighth  term  of 


(^^-^r* 


THE  BtNTOMIAL  THEOREM.  329 

394.    To  find  any  root  of  a  number  approximately  by  the 
Binomial  Theorem. 

1.   Find  the  approximate  value  of  -^2^  to  five  places  of 
decimals. 

^25  =  25*  =  (27  -  2)^  =  (3^  -  2)*. 

Expanding  b}'  the  Binomial  Theorem,  we  have 

[(3^)  +  (-2)]'  =  (3^)^  +  |(30-*(-2)-i(3r^(-2)« 

+  ^(3r^(-2)«-- 

^g__2 £_  40 


3.32      9.3^      81-3« 

Expressing  the  value  of  each  fraction  approximately  to  five 
places  of  decimals,  we  have 

^25  =  3  -  .07407  -  .00183  -  .00008 

=  2.92402,   Ans. 

RULE. 

Separate  the  given  number  into  two  parts,  the  first  of  tvhich 
is  the  nearest  perfect  power  of  the  same  degree  as  the  required 
root. 

Expand  the  result  by  the  Binomial  Theorem. 

Note.  If  the  second  term  of  the  binomial  is  small  compared  with 
the  first,  the  terms  of  the  expansion  diminish  rapidly;  but  if  the  second 
term  is  large  compared  with  the  first,  it  requires  a  great  many  terms  to 
ensure  any  degree  of  accuracy. 

EXAMPLES. 

Find  the  approximate  values  of  the  following  to  five  places 
of  decimals : 

2.  VIO-  *•    V^'  6.    ^17. 

3.  V^^-  5-    </''^-  '^'    ■\/2S' 


330  ALGEBRA. 


XXXVI.     LOGARITHMS. 

395.  Every  positive  number  may  t»e  expressed,  exactly  or 
approximate!}',  as  a  power  of  10  ;  thus, 

100=102;  13  =  lO^"^- ;  etc. 

When  thus  expressed,  the  corresponding  exponent  is  called 
its  Logarithm  to  the  base  10;  thus,  2  is  the  logarithm  of  100 
to  the  base  10,  a  relation  which  is  written 

logio  100  =  2,  or  simply  log  100  =  2. 

And  in  general,  if  10*  =  m,  then  x  =  logm. 

396.  Any  positive  number  except  unity  may  be  taken  as 
ihe  base  of  a  system  of  logarithms;  thus,  if  0^=171,  then 
a;  =  log„m. 

Logarithms  to  the  base  10  are  called  Common  Logarithms, 
and  are  the  only  ones  used  for  numerical  computations. 
If  no  base  is  expressed,  the  base  10  is  understood. 

397.  By  Arts.  220  and  221,  we  have  . 

100=1,  10-1  =  —     ^1 

10 

10^=10,  10-2  =  J_    =.01, 

100 

102=100,  10-3  ==_1_=:  .001,  etc. 

1000 

Whence,  by  the  definition  of  Art.  395, 

log  1  =  0,  log.l  =-1  =  9-10, 

log  10  =  1,  log  .01  =.  -  2  =  8  -^  10, 

log  100  =  2,  log  .001  =  -3  =  7-10,  etc. 

Note.  The  second  form  of  the  results  for  log  .1,  log  .01,  etc.,  is  pref- 
erable in  practice.  In  each  of  the  last  six  equations  the  base  10  is 
understood  (Art.  396). 


LOGARITHMS.  331 

398.  It  is  evident  from  Art.  397  that  the  logarithm  of  a 
number  greater  than  1  is  positive,  and  that  the  logarithm  of 
a  number  between  0  and  1  is  negative. 

399.  If  a  number  is  not  an  exact  power  of  10,  its  common 
logarithm  can  only  be  expressed  approximately ;  the  integral 
part  of  the  logarithm  is  called  the  characteristic,  and  the 
decimal  part  the  mantissa. 

For  example,  log  13=  1.1139. 

In  this  case  the  characteristic  of  the  logarithm  is  1,  and 
the  mantissa  is  .1139. 

400.  It  is  evident  from  the  first  column  of  Art.  397  that 
the  logarithm  of  any  number  between 

1  and      10  is  equal  to  0  plus  a  decimal ; 
10  and    100  is  equal  to  1  plus  a  decimal ; 
100  and  1000  is  equal  to  2  plus  a  decimal ;  etc. 

Hence,  the  characteristic  of  the  logarithm  of  a  number 
with  one  figure  to  the  left  of  its  decimal  point,  is  0;  with  two 
figures  to  the  left  of  the  decimal  point,  is  1  ;  with  three  figures 
to  the  left  of  the  decimal  point,  is  2  ;  etc. 

401.  In  like  manner,  from  the  second  column  of  Art.  397, 
the  logarithm  of  a  decimal  between 

1  and      .1  is  equal  to  9  plus  a  decimal  —  10  ; 
.1  and    .01  is  equal  to  8  plus  a  decimal  —  10  ; 
.01  and  .001  is  equal  to  7  plus  a  decimal  —  10 ;  etc. 

Hence,  the  characteristic  of  the  logarithm  of  a  decimal 
with  no  ciphers  between  its  decimal  point  and  first  significant 
figure,  is  9,  with  —10  after  tlie  mantissa ;  of  a  decimal  with 
o)ie  cipher  between  its  point  and  first  figure  is  8,  with  —10 
after  the  mantissa ;  of  a  decimal  with  two  ciphers  between 
its  point  and  first  figure,  is  7,  with  —10  after  the  mantissa; 
etc. 


332  ALGEBRA. 

402.  For  reasons  which  will  be  given  hereafter,  only  the 
mantissa  of  the  logarithm  is  given  in  a  table  of  logarithms  of 
numbers  ;  the  characteristic  must  be  supplied  by  the  reader. 

The  rules  for  characteristic  are  based  on  Arts.  400  and  401  : 

I.    If  the  number  is  greater  than  1,  the  characteristic  is  1 

less  than  the  number  of  places  to  the  left  of  the  decimal  point. 

II.    If  the  number  is  less  than  1,  subtract^  the  mimber  of 

ciphers  between  the  decimal  point  and  first  significant  figure 

from  9,  writing  —10  after  the  mantissa. 

Thus,  characteristic  of  log  906328.5  =  5  ; 

characteristic  of  log    .007023  =  7,  with  — 10  after 
the  mantissa. 

Note.  Some  writers,  in  dealing  with  the  characteristics  of  negative 
logarithms,  combine  the  two  portions  of  the  characteristic,  and  write 
the  result  as  a  negative  characteristic  before  the  mantissa. 

Thus,  instead  of  7.6036  -  10,  the  student  will  frequently  find  3.6036, 
a  minus  sign  being  written  over  the  characteristic  to  denote  that  it 
alone  is  negative,  the  mantissa  being  always  positive. 

PROPERTIES    OF    LOGARITHMS. 

403.  In  any  system,  the  logarithm  of  unity  is  zero. 
For  since  a^  =1,  we  have  log^l  =  0  (Art.  395). 

404.  In  any  system,  the  logarithm  of  the  base  itself  is  unity. 
For  since  a^  =  a,  we  have  log^a  =  1. 

405.  In  any  system  whose  base  is  greater  than  unity,  the 
logarithm  of  zero  is  minus  infinity. 

1        1 
For  if  a  is  >1,  we  have  a"^  =  —  =  -=  0  (Art.  300). 

'  ^00  GO  ^  ^ 

Whence  by  Art.  395,  log^O  =  —  oo. 

Note.  As  stated  in  Art.  301,  no  literal  meaning  can  be  attached  to 
the  result  loga  0  =  —  oo  ;  it  must  be  interpreted  as  indicated  in  Art.  300. 

That  is,  if  in  any  system  whose  base  is  greater  than  unity,  a  number 
approaches  zero  as  a  limit,  its  logarithm  is  negative,  and  increase* 
without  limit  in  absolute  value. 


LOGARITHMS.  333 

406.  In  any  system,  the  logarithm  of  a  product  is  equal  to 
the  sum  of  the  logarithms  of  its  factors. 

Assume  the  equations 

«^  ='''];  whence,  by  Art.  395,  |  ^  =  ^^^«^^' 
a^=n  )  (  y  =  log„ri. 

Multiplying,  we  have 

a' X  a^  =  m7i,  or  a' '^^  =  mn. 

Whence,         logo77i?i  =  a^-f^/- 
Substituting  the  values  of  x  and  y^  we  have 

log„mn  =  log„m  -|-  loga?i. 
In  like  manner,  the  theorem  may  be  proved  for  the  product 
of  three  or  more  factors. 

407.  By  aid  of  the  theorem  of  Art.  406,  the  logarithm  of 
any  composite  number  may  be  found  -when  the  logarithms 
of  its  factors  are  known. 

1.    Given  log  2  =  .3010,  and  log3  =  .4771  ;  find  log72. 

log72  =  log(2x2x  2x3x3) 

=  log2  +log2  +  log2  +log3  +log3 
=  3xlog2  +  2  xlog3 
=  .9030  +  .9542 
=  1.8572,  Ans. 

EXAMPLES. 

Given  log2  =  .3010,  log3  =  .4771,  log5  =  .6990,  and 
log 7  =  .8451  ;  find: 

2.  log  21.  7.  log  98.  12.  log  135.  17.  log  1134. 

3.  log  63.  8.  log  105.  13.  log  168.  18.  log  5145. 

4.  log  56.  9.  log  112.  14.  log  147.  19.  log  7056. 

5.  log  84.  10.  log  144.  15.  log  375.  20.  log  14406. 

6.  log  45.  11.  log  216.  16.  log  343.  21.  log  15552. 


834  ALGEBRA. 

408.  In  any  system^  the  logarithm  of  a  fraction  is  equal  to 
the  logarithm  of  the  iiumerator  minus  the  logarithm  of  the 
denominator. 

Assume  the  equations 

""  =  ™|;  whence,  j'"  =  '<'§« »»' 
a^  —  n  )  ^y—  log„n. 

Dividing,  we  have  —  =  — ,  or  a*  ^  =  — 
a^      n  n 

Whence,  log^  ~  =  x  —  y. 

Substituting  the  values  of  x  and  y, 

m 
log,-  =  log«m-log„7i. 


409.    1.  Given  log  2  =.3010;  find  log  5. 
log  10  — log  2 
1-. 3010  =.6990,  Ans. 


log  5  =  log  —  =  log  10  -  log  2 


EXAMPLES. 
Given  log  2  =  . 3010,  log  3=. 4771,  and  log  7  =  .8451 ;  find  : 

2.  log-.  5.   log 35.  8.   log—.  11.   log7f 

3.  log  12.  6.   log—.  9.   log  175.        12.   log—. 

^7  ^16  *  °  6 

4.  log3f  7.   log  125.        10.   log  Hi.        13.   log5f 

410.    In  any  system^  the  logarithm  of  any  power  of  a  quan- 
tity is  equal  to  the  logarithm  of  the  quantity  multiplied  hy  the 
exponent  of  the  power.     . 
Assume  the  equation 

a''  =  m\  whence,  x  =  log„ m. 
Raising  both  members  to  the  pth  power,  we  have 

a^*  =  m^  ;  whence,  log„  m^  =px=p  log^  m. 


LOGARITHMS.  335 

411.  In  any  system^  the  logarithm  of  any  root  of  a  quantity 
is  equal  to  the  logarithm  of  the  quantity  divided  by  the  index  of 
the  root. 

I       1 
For,  loga  -{/m  =  log«(m'")  =  -  log^  m  (Art.  410) . 

412.  1.  Given  log  2  =  .3010  ;  find  the  logarithm  of  2i 

log  2^  =  -  X  log  2  =  5  X  .3010  =  .5017,  Ans. 
o  o 

Note.  To  multiply  a  logarithm  by  a  fraction,  multiply  first  by  the 
numerator,  and  divide  the  result  by  the  denominator. 

2.    Given  log3  =  .4771  ;  find  the  logarithm  of  ^3. 

log  ^3=1^=  '^^  =  .0596,  Ans, 


EXAMPLES. 
Given  log 2=. 3010,  log3  =  .4771,  and  log 7  =  .8451 ;  find: 

3.  log3i  7.  logl2i  11.  logl5i  15.  log ^5. 

4.  log2^  8.  log2li  12.  logV7.  16.  log  ^35. 
6.  log7^  9.  logl4\  13.  log</3.  17.  log  ^98. 
6.  log5i  10.  log25i  14.  log  ^2.  18.  log  ^126. 

19.    Find  the  logarithm  of  (2^  x  3^) . 

By  Art.  406,  log  (2^  x  3^)  =  log  2^  -f  log  3^ 

=  41og2  +  |log3 
=  .1003  +  .5964  =  .6967,  Ans. 

Find  the  values  of  the  following  : 

20.  logQ!  22.1og(3^x2t).  24.   log^lj.  26.   log^y- 

21.  log^.       23.   log3-^7.        25.   log:f^.  27.   log^. 

5f  ,  V^  10^ 


336  '  ALGEBRA. 

413.  In  the  common  system^  the  mantissce  of  the  logarithms 
of  numbers  having  the  same  sequence  of  figures  are  equal. 

To  illustrate,  suppose  that  log  3.053  =  .4847  ;  then, 

log  30.53    =  log  (10x3.053)    =  log  10  +  log  3.053 

=  l  +  .4847  =1.4847; 

log  305.3    =  log  (100  X  3.053)  =  log  100  +  log3.053 

=  2 +  .4847  =2.4847; 

log  .03053  =  log  (.01  X  3.053)  =  log  .01  +  log  3.053 

=  8  -  10  +  .4847      =  8.4847  -  10  ;  etc. 

It  is  evident  from  the  above  that  if  a  number  is  multiplied 
or  divided  by  any  integral  power  of  10,  producing  another 
number  with  the  same  sequence  of  figures,  the  mantissae  of 
their  logarithms  will  be  equal. 

Thus,  if  log  3.053  =  .4847,  then 

log  30.53  =  1.4847,         log  .3053      =  9.4847  -  10, 

log  305.3  =  2.4847,         log  .03053    =8.4847-10, 

log  3053.  =  3.4847,         log  .003053  =  7.4847  -  10,  etc. 

Note.  The  reason  will  now  be  seen  for  the  statement  made  in 
Art.  402,  that  only  the  mantissas  are  given  in  a  table  of  logarithms  of 
numbers.  For,  to  find  the  logarithm  of  any  number,  we  have  only  to 
take  from  the  table  the  mantissa  corresponding  to  its  sequence  of 
figures,  and  the  characteristic  may  then  be  prefixed  in  accordance  with 
the  rules  of  Art.  402, 

This  property  of  logarithms  is  only  enjoyed  by  the  common  system, 
and  constitutes  its  superiority  over  others  for  the  purposes  of  numeri- 
cal computation. 

414.  1.  Given  log 2  =  .3010,  log 3  =  .4771;  find  log. 00432. 

log  432  =  log  (2^  X  3=^)  =  4  log  2  +  3  log  3 
=  1.2040  +  1.4313  =  2.6353. 
Then  by  Art.  413,  the  mantissa  of  the  result  is  .6353. 
Whence  by  Art.  402,  log  .00432  =  7.6353  -  10,  Ans. 


LOGARITHMS.  337 


EXAMPLES. 

Given    log  2  =.3010,    log  3  =  .4771,    and    log  7  =.8451; 
find: 

2.  log  1.8.  7.  log  .0054.  12.  log  302.4. 

3.  log  2.25.  8.  log  .000315.  13.  log  .06174. 

4.  log. 196.  9.  log  7350.  14.  log(8.1)^ 
6.  log  .048.  10.  log  4.05.  15.  log  </9T6'. 
6.  log38.4.  11.  log  .448.  16.  log(22.4)i 

415.    To  prove  the  relation 

log6m=  .  "\' 
Assume  the  equations 

«'  =  »!;  whence,  I  *=;°S»'''' 
b^  =  m)  Ly  =  \ogi7n. 

From  the  assumed  equations,  we  have 

a'  =  b%  or  a*  =  b. 
Whence,  loga6  =  -,    or  y  = 


y  iog«& 

Substituting  the  values  of  x  and  ?/, 
loge,m  =  -^^. 

By  aid  of  this  relation,  if  the  logarithm  of  a  quantity  m  to 
a  certain  base  a  is  known,  its  logarithm  to  any  other  base  b 
may  be  found  by  dividing  by  the  logarithm  of  b  to  the  base  a. 

416.    To  prove  the  relation 

logftaxlog„6  =  l. 
Putting  m  =  a  in  the  result  of  Art.  415,  we  have 

log,a  =  |^  =  -J-   (Art.  404). 
log„6      log„6 

Whence,  log^a  x  log«6  =  1. 


338 


ALGEBRA. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

II 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

"39 

"73 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

'2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

37" 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5"9 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

55«^2 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

59" 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7335 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

|No. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

LOGARITHMS. 


339 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

?^i9 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

'^'ll 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

917s 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

340  ALGEBRA. 

USE  OF  THE  TABLE. 

417.  The  table  (pages  338  and  339)  gives  the  mantissse 
of  the  logarithms  of  all  integers  from  100  to  1000,  calcu- 
lated to  four  places  of  decimals. 

418.  To  find  the  logarithm  of  a  number  of  three  figures. 

Find  in  the  column  headed  "  No."  the  first  two  significant 
figures  of  the  given  number. 

Then  the  mantissa  required  will  be  found  in  the  corre- 
sponding horizontal  line,  in  the  vertical  column  headed  by 
the  third  figure  of  the  number. 

Finally,  prefix  the  characteristic  by  the  rules  of  Art.  402. 

For  example,         log  168  =  2.2253  ; 

log  .344  =  9.5366  -  10  ;  etc. 

419.  For  a  number  consisting  of  one  or  two  significant 
figures,  the  column  headed  0  may  be  used. 

Thus,  let  it  be  required  to  find  log  83  and  log  9. 
By  Art.  413,  log  83  has  the  same  mantissa  as  log  830,  and 
log  9  the  same  mantissa  as  log  900.     Hence, 

log  83  =  1.9191,  and  log  9  =  0.9542. 

420.  To  find  the  logarithm  of  a  number  of  more  than  three 
figures. 

1.    Required  the  logarithm  of  327.6. 
We  find  from  the  table,  log  327  =  2.5145, 
log  328  =  2.5159. 

That  is,  an  increase  of  one  unit  in  the  number  produces  an 
increase  of  .0014  in  the  logarithm. 

Therefore  an  increase  of  .6  of  a  unit  in  the  number  will 
produce  an  increase  of  .6  x  .0014  in  the  logarithm,  or  .0008 
to  the  nearest  fourth  decimal  place. 

Hence,     log  327.6  =  2.5145  +  .0008  =  2.5153. 


LOGARITHMS.  341 

Note.  The  difference  between  any  mantissa  in  the  table  and  the 
mantissa  of  the  next  higher  number  of  three  figures,  is  called  the  tabu- 
lar difference.     The  subtraction  may  be  performed  mentally. 

The  following  rule  is  derived  from  the  above  : 

Find  from  the  table  the  mantissa  of  the  first  three  significant 
figures,  and  the  tabular  difference. 

Multiply  the  latter  by  the  remaining  figures  q/  the  number, 
with  a  decimal  point  before  them. 

Add  the  result  to  the  mantissa  of  the  first  three  figures,  and 
prefix  the  proper  characteristic. 


EXAMPLES. 
2.    Find  the  logarithm  of  .021508. 


Mantissa  of  215  =  3324 

Tabular  difference  =     21 

2 

.08 

3326 

Correction  =  1.68  = 

:  2,  nearly. 

Ans.    8.3326-  10. 

Find  the  logarithms  of  the  following : 

3.    80.          7.    .7723. 

11.    20.08.           15.    5.1809. 

4.    6.3.         8.    1056. 

12.    92461.          16.    1036.5. 

5.    298.        9.    3.294. 

13.    .40322.         17.    .086676. 

6.    .902.     10.    .05205. 

14.    .007178.       18.    .11507. 

421.    To  find  the  number  corresponding  to  a  logarithm. 
1.    Required  the  number  whose  logarithm  is  1.6571. 
Find  in  the  table  the  mantissa  6571. 

In  the  corresponding  line,  in  the  column  headed  "No.,' 
we  find  45,  the  first  two  figures  of  the  required  number,  and 
at  the  head  of  the  column  we  find  4,  the  third  figure. 

Since  the  characteristic  is  1,  there  must  be  two  figures  to 
the  left  of  the  decimal  point  (Art.  402) . 

Hence,  number  corresponding  to  1.6571  ^  45,4,  4n$, 


342  ALGEBRA. 

2.  Required  the  number  whose  logarithm  is  2.3934. 

We  find  in  the  table  the  mantissas  3927  and  3945,  whose 
corresponding  numbers  are  247  and  248,  respectively. 

That  is,  an  increase  of  18  in  the  mantissa  produces  an 
increase  of  one  unit  in  the  number  corresponding. 

Therefore,  an  increase  of  7  in  the  mantissa  will  produce 
an  increase  of  ^-^  of  a  unit  in  the  number,  or  .39,  nearly. 

Hence,  number  corresponding  =  24 7  +  . 39  =  24 7. 39,  Ans. 

The  following  rule  is  derived  from  the  above  : 

Find  from  the  table  the  next  less  mantissa,  the  three  figures 
corresponding,  and  the  tabular  difference. 

Subtract  the  next  less  from  the  given  mantissa,  and  divide 
the  remainder  by  the  tabular  difference. 

Annex  the  quotient  to  the  first  three  figures  of  the  number, 
and  point  off  the  result. 

Note.  The  rules  for  pointing  off  are  the  reverse  of  those  of  Art.  402  : 
I.  7/*— 10  is  not  written  after  the  mantissa,  add  1  to  the  characteristic ^ 
giving  the  number  of  places  to  the  left  of  the  decimal  point. 

II.  If—IQ  is  written  after  the  mantissa,  subtract  the  positive  part  of  the 
characteristic  from  9,  giving  the  number  of  ciphers  between  the  decimal  point 
and  first  significant  figure. 

EXAMPLES. 

3.  Find  the  number  whose  logarithm  is  8.5264  —  10. 

5264 
Next  less  mantissa =5263  ;  three  figures  corresponding =336. 
Tabular  difference  =  13)  1.000(. 077  =  .08,  nearly. 

91 
90 

According  to  the  above  rule,  there  will  be  one  cipher  be- 
tween the  decimal  point  and  first  significant  figure. 

Hence,  number  corresponding  =  .033608,  Ans, 


LOGARITHMS.  343 

Find  the  numbers  corresponding  to  the  following  loga- 
rithms : 

4.  1.8055.  9.  8.1648-10.  14.  1.6482. 

5.  9.4487-10.  10.  7.5209-10.  15.  7.0450-10. 

6.  0.2165.  11.  4.0095.  16.  4.8016. 

7.  3.9487.  12.  0.9774.  17.  8.1144-10. 

8.  2.7364.  13.  9.3178-10.  18.  2.7015. 

APPLICATIONS. 

422.  The  value  of  an  arithmetical  quantity,  in  which  the 
operations  indicated  involve  only  multiplication,  division, 
involution,  or  evolution,  may  be  most  conveniently  found  by 
logarithms. 

The  utility  of  the  process  consists  in  the  fact  that  addition 
takes  the  place  of  multiplication,  subtraction  of  division, 
multiplication  of  involution,  and  division  of  evolution. 

Note.  In  computations  with  four-place  logarithms,  the  results  can- 
not usually  be  depended  upon  to  more  than /our  significant  figures. 

423.  1.    Find  the  value  of  .0631  x  7.208  x  .51272. 

By  Art.  406,  log  (.0631  x  7.208  x  .51272) 

=  log  .0631  +  log  7.208  -f-  log  .51272. 
log    .0631  =    8.8000  -  10 
log    7.208=    0.8578 
W  .51272=    9.7099-10 


Adding,  .-.  log  of  result  =  19.3677  -  20 

=    9.3677-10  (see  Note  1) 

Number  corresponding  to  9.3677  —  10  =  .2332,  Arts. 

Note  1.   If  the  sum  is  a  negative  logarithm,  it  should  be  reduced  so 
that  the  negative  portion  of  the  characteristic  may  be  — 10. 
Thus,  19.3677-20  is  reduced  to  9.3677-10, 


344  ALGEBRA. 

2.  P^ind  the  value  of  5?^. 

7984 

By  Art.  408,  log  ?5M  =  w  336.8  -  W  7984. 

7984  ^  ^ 

log  336.8  =  12.5273  -  10     (see  Note  2) 
log  7984=    3.9022 
Subtracting,  .-.  log  of  result  =    8.6251  —  10 

Number  corresponding  =  .04218,  Ans. 

Note  2.  To  subtract  a  greater  logarithm  from  a  less,  or  to  subtract 
a  negative  logarithm  from  a  positive,  increase  the  characteristic  of  the 
minuend  by  10,  writing  — 10  after  the  mantissa  to  compensate. 

Thus,  to  subtract  3.9022  from  2.5273,  write  the  minuend  in  the  form 
12.5273  - 10 ;  subtracting  3.9022  from  this,  the  result  is  8.6251  -  10. 

3.  Find  the  value  of  (.07396)^ 

By  Art.  410,  log  (.07396)^  =  5  x  log  .07396. 

log  .07396  =  8.8690 -10 
5 


44.3450  -  50 
=  4.3450-10   (see  Note  1) 
=  log  .000002213,  Ans. 


4.    Find  the  value  of  V. 035063. 


By  Art.  411,  log  V. 035063  =  -log  .035063. 

o 

log  .035063  =  8.5449 -10 

20.  -  20  (see  Note  3) 

9.5150-10 
=  log  .3274,  Ans. 

Note  3.  To  divide  a  negative  logarithm,  add  to  both  parts  such  a 
multiple  of  10  as  will  make  the  negative  portion  of  the  characteristic 
exactly  divisible  by  the  divisor,  with  —10  as  the  quotient. 

Thus,  to  divide  8.5449  — 10  by  8,  add  20  to  both  parts  of  the  loga- 
rithm, giving  the  result  28.5449—  30,  Dividing  this  by  3,  the  quotient 
18  9,6160-10, 


LOGARITHMS.  346 


ARITHMETICAL    COMPLEMENT. 

424.  The  Arithmetical  Complement  of  the  logarithm  of  a 
Dumber,  or,  briefly,  the  CologaritJim  of  the  number,  is  the 
logarithm  of  the  reciprocal  of  that  number. 

Thus,  colog  409  =  W—  =  log  1  -  log  409. 

log  1=10.  -  10     (Note  2,  Art.  423) 

log  409  =   2.6117 
.•.colog409=    7.3883-10. 

Again,  colog  .067  =  log ;:;  =  log  1  —  log  .067. 

.067 

log  1  =  10.         -10 
log  .067  =    8.8261-10 
.-.colog  .067=    1.1739 
The  following  rule  is  evident  from  the  above : 

To  find  the  cologanthm  of  a  number,  subtract  its  logarithm 
from  10-10. 

Note.  The  cologarithm  may  be  obtained  from  the  logarithm  by 
subtracting  the  last  significant  figure  from  10  and  each  of  the  others 
from  9,  —  10  being  written  after  the  result  in  the  case  of  a  positive 
logarithm. 

425.  Example.    Find  the  value  of  —^' 


8.709  X  .0946 


,  .51384  ,        f  K^ooA  1  1 

log =  log(. 51384  X 


8.709  X  .0946         °V  ^•'^^^      .0946/ 

=  log  .51384  +  log  ^^^  +  log     ^ 


8.709         °.0946 
=  log  .51384  +  colog  8.709  +  colog  .0946. 
log  .51384  =  9.7109 -10 
colog  8.709  =  9.0601 -10 
colog  .0946  =  1.0241 

9.7951 -10  =  log  .6239,  Ans. 


346  ALGEBRA. 

It  is  evident  from  the  above  that  the  logarithm  of  a  fraction 
is  equal  to  the  logarithm  of  the  numerator  plus  the  eologa- 
rithm  of  the  denominator. 

Or  in  general,  to  find  the  logarithm  of  a  fi-action  whose 
terms  are  composed  of  factors, 

Add  together  the  logarithms  of  the  factors  of  the  7iumerator^ 
a7id  the  cologarithms  of  the  factors  of  the  denommator. 

Note.  The  value  of  the  above  fraction  may  be  found  without  using 
cologarithms,  by  the  following  formula  : 

log ^l^§i —  =  log  .51384  -  log  (8.709  X  .0946) 

^  8.709  X. 0946        ^  &\  ^  J 

=  log  .51 384  -  (log  8 .  709  +  log  .0946) . 
The  advantage  in  the  use  of  cologarithms  is  that  the  written  work 
of  computation  is  exhibited  in  a  more  compact  form. 


EXAMPLES. 

426.  Note.  A  negative  quantity  can  have  no  common  logarithm, 
as  is  evident  from  the  definition  of  Art.  395.  If  negative  quantities 
occur  in  computation,  they  may  be  treated  as  if  they  were  positive,  and 
the  sign  of  the  result  determined  irrespective  of  the  logarithmic  work. 

Thus,  in  Ex.  3,  p.  346,  the  value  of  721.3  x  (-3.0528)  may  be  ob- 
tained by  finding  the  value  of  721,3x3.0528,  and  putting  a  negative 
sign  before  the  result.     See  also  Ex.  34,  p.  347. 

Find  by  logarithms  the  values  of  the  following : 

1.  9.238X.9152.  4.    (- 4.3264)  x  (- .050377). 

2.  130.36  x  .08237.  5.    .27031  x  .042809. 

3.  721.3  x(- 3.0528).       6.    (- .063165)  x  11.134. 

7     401.8  g     -  .3384  jj     22518 

52.37*  *     .08659  *  *    64327* 

g     7.2321  jQ        9.163  ^g       .007514 


10.813  .0051422  -.015822 

13  3.3681  ^^     15.008  X  (-.0843) 

12.853  X  .6349*  '        .06376  X  4.248 


LOGARITHMS.  347 

15     (-2563)  X. 03442 
714.8  X  (-.511)  ' 

jg  121.6  X  (-9.025) 

(-48.3)  X  3662  X  (-.0856)* 


17.  (23.86)^  22.  (.8)^  28.  V.4294. 

18.  (.532)8.  23.  (-3.16)i  29.  ^.02305. 

19.  (-1.0246)^       24.  (.021)i  30.  -v^lOOO. 

20.  (.09323)*.  25.  ^2.  31.  a/- .00951. 

21.  5^.  26.  </5.  32.  ^.0001011. 

27.  V^^. 

33.   Find  the  value  of  ^^. 
3^ 

log  ^^  =  log  2  +  log  ^'5  +  colog  3^  (Art.  425) 
3^ 

=  log2  +  ilog5  4-|colog3. 

log  2=    .3010 
log  5  =    .6990  ;  divide  by  3  =    .2330 

colog  3  =  9.5229 -10;  multiply  by  |=  9.6024  -  10 

.1364 
=  log  1.369,  Ans, 


34.    Find  the  value  of  \i—^^—— — 


^ 


^^^  4fiM = *  ^""^  fSi = ^  ^^^^  '^^^^^ "  ^^^  ^'^^^^ 


log  .03296=    8.5180-10 
log  7.962    =    0.9010  " 

3)27.6170-30 


9. 2057 -10  =  log. 1606. 

Ans.  -.1606. 


348  ALGEBRA. 

Find  the  values  of  the  following : 


35.    22x3^ 
3t 


36 


37. 


38. 


39.    f-^^ 


40. 


41.     «l?i. 


Y^. 


.08726X1 
.1321 j 

n 

13' 


57- 


(-10) 


42. 


43. 


3 


\3 


-€ 


45 
46. 
47. 

48. 


■^l 


3258 


49309 
31.63\T'f 


429 

100^ 

(.7325)^ 

3/ 


V. 0001289 

V'.0008276 


50. 


51. 


VI 13, 
VTOO73 


44.    ^2  X  ^3  X  ^:or.     49.    (n^IML^ 

-  (.2345)^ 


(.68291)^ 
V5.955  X  Vei^ 

-v/298.54 


63.    (18.9503)11  X  (-.1)". 


54.    V3734.9  x  .00001108. 


52.    (538.2  X  .0005969)^.  55.    (2.6317)*  x  (.71272)i 

gg     -^-.008193  X  (.06285)^ 
-  .98342 

57.    V^035  X  \/.62667  x  a/.0072103. 


EXPONENTIAL  EQUATIONS. 

427.  An  Exponential  Equation  is  one  in  which  the  un- 
known quantity  occurs  as  an  exponent. 

To  solve  an  equation  of  this  form,  take  the  logarithms  of 
both  members ;  the  result  will  be  an  equation  which  can  be 
solved  b}'  ordinary  algebraic  methods. 

1.    Given  3P  =  23  ;  find  the  value  of  x. 

Taking  the  logarithms  of  both  members, 

log(3P)  =  log23. 


LOGARITHMS.  349 

Or,  by  Art.  410, 

a;log31  =  log23. 

Whence,  x  =  1^^  =  h^^  =  .91303,  Arts, 

log  31      1.4914 

2.    Given  .2='  =  3  ;  find  the  value  of  x. 

Taking  the  logarithms  of  both  members, 

a; log  .2  =log3. 

wu  log  3  .4771  .4771 

Whence,  .  =  ^  =  ^^^^^^-^^  = -^^ 

=  -.6825,  Ans. 

EXAMPLES. 
Solve  the  following  equations  : 

3.  IP  =3.  5.    13^  =.281.  7.    a^  =  6"*c". 

4.  .3=^  =.8.  6.    .703^=1.096.         8.   ma''  =  n. 
9.    21^-2.^9260.  10.    .051^+^=384.4. 

11 .  Given  a,  ?%  and  I ;  derive  the  formula  for  n.  (Art.  350.) 

12.  Given  a,  r,  and  S  ;  derive  the  formula  for  n. 

13.  Given  a,  Z,  and  ^S  ;  derive  the  formula  for  n. 

14.  Given  r,  Z,  and  S  ;  derive  the  formula  for  n. 

15.  Find  the  logarithm  of  .3  to  the  base  7. 

By  Art.  415,  we  have 

log,  .3  =  l^H^  =  9:iIZi^  =  _  .522?  ^  _  ^^^^^ 

^'  logio7  .8451  .8451 

Find  the  values  of  the  following : 

16.  logall.  18.    log.365.  20.   log7356.31. 

17.  logs  .8.  19.   log.8.0823.        21.   log^^  .007228. 


360 


ALGEBRA. 


EXPONENTIAL  AND  LOGARITHMIC   SERIES. 
428.    We  have  for  all  values  of  n  and  x^ 

1  + 


nj  \        n 


Expanding  both  members  by  the  Binomial  Theorem, 


[-; 


n{n-l)  I       n(n-l){n-2)  I 


+  •- 


I 


\2        n^  [3 

r=l   I  nx^   I  ^^O^^-l)  1    I  nx{nx-l){nx-2)  1 
w  [2  n^  [3  71^ 

That  is, 


1  +  1  + 


1- 


1  - 


l^ 


^ 


+  •• 


=  l+a^  + 


x\  X 


i) 


+ 


This  equation  holds  however  great  n  may  be. 
Now  let  n  be  indefinitelv  increased. 


(1) 


Then  since  each  of  the  terms 


n    n 


,  approaches  the 


]• 


limit  0  (Art.  301),  the  limit  of  the  first  member  of  (1)  ia 
11  +  1  +  -  +  ^  +  .. 

and  the  limit  of  the  second  member  is 
l+i»  +  -  +  -  +  -- 

By  the  Theorem  of  Limits    (Art.  299)  these   limits  are 
equal ;  that  is. 


l  +  l+,4  +  ^  + 


^+^+1+1+ 


[2      ^ 
Denoting  the  series  in  brackets  b}-  e,  we  obtain 

e*  =  l+a;  +  .-+-  +  ... 
[2      |3 


(2) 


LOGARITHMS.  351 

429.  Substituting  ma;  in  place  of  x  in  (2),  Art.  428,  we 

^'"^  =  1+^^  +  ^+^  +  -  (3) 

Let  e"*  =  a  ;  then  m  =  log«a  (Art.  395) ,  and  e"**  =  a*. 
Substituting  in  (3) ,  we  obtain 

a«  =  1  4-  (logea) X  +  (log.a)^^  +  (logea)^ ^  +  ...      (4) 
This  result  is  called  the  Exponential  Theorem. 

430.  The  system  of  logarithms  which  has  e  for  its  base 
is  called  the  Napienan  System,  from  Napier,  the  inventor  of 
logarithms. 

The  approximate  value  of  e  may  be  readily  calculated  by 
aid  of  the  series  of  Art.  428, 

e=l+l+-+-+-+— 

{^     \l     a 

and  will  be  found  to  equal  2,7182818... 

431.  To  expand  log.  {\-\-x)  m  ascending  powers  of  x. 

Substituting  in  equation  (4),  Art.  429,  1  -f-  a;  in  place  of  a, 
and  y  in  place  of  ic,  we  obtain 

(1 +  «)"=!+  [log,(lH-aj)]2/  +  termsin^S  f,  etc. 

Expanding  the  first  member  by  the  Binomial  Theorem, 

\2  [3 

=  1  +  [log,(l -\-x)'\y  +  terms  in  2/^  2/^,  etc. 
This  equation  holds  for  every  value  of  y  which  makes  both 
members  convergent,  and  by  the  Theorem  of  Undetermined 
Coefficients  the  coefficients  of  y  in  the  two  series  are  equal. 

11         12         13 
That  is,  aj-'^ar^  +  ^a^^-^a^'H- --105,(1  + a;); 

L^       1^       il 

n^  /v3  /y»4 

or,  log.(l  +  a;)  =  a;-^  +  ---  +  ... 

»  oA  -r    y  2       3       4 


352  ALGEBRA. 

432.  The  above  formula  can  be  used  for  the  calculation 
of  Napierian  logarithms  if  x  is  so  taken  that  the  series  in 
the  second  member  is  convergent ;  but  unless  x  is  small,  it 
requires  the  sum  of  a  great  many  terras  to  ensure  any  degree 
of  accuracy. 

433.  To  derive  a  more  convenient  formula  for  calculating 
the  Napierian  logarithm  of  a  number. 

By  Art.  431,  we  have 

/y»2  /Y"  /V"  /V" 

loge(l  +  a:)=      x-~-^----{~- (1) 

&ev    -r    y  2       3       4       5  ^ 

Putting  —  a;  in  place  of  x,  this  becomes 

/y»2  /y.3  nA  /ywS 

"''^  ^  2       3       4       5  ^  ^ 

Subtracting  (2)  from  (1),  we  have 

loge(14-a^)-log.(l-aj)  =  2a;-h2|  +  2|  +  ... 

Whence,  by  Art.  408, 

log.i±|  =  2(.  +  f  +  f  +  ...).  (3) 

m  —  n 

m  —  n^       \-\-x  m-\-n      2m      m 

Let       x  = ; — ;  then = =  ir"  =  — 

m  -\-n  \  —  X  m  —  71      2n       n 

m-\-n 
Substituting  these  values  in  (3),  we  obtain 

log.™  =  2  r???^^  + 1  ("H^^^  1  (^2!?i::i^Y+  •  •  -I- 

n        \jn  -\-n      3  \m  -^nj      5  \m  +nj  J 

But  by  Art.  408,  log^  —  =  logem  —  log^n. 
n 

Whence, 

,               1           i  of^  — w  ,  1/m  — n\^,  \fm  —  n\.       ~| 
log,m  =  log,n  +  2   r— +  ^   r"-    +n   T"    + '•'   ' 

434.  Let  it  be  required,  for  example,  to  calculate  the 
Napierian  logarithm  of  2  to  six  places  of  decimals. 


LOGARITHMS.  353 

Putting  m  =  2  and  n  ==  1  in  the  result  of  Art.  433,  we  have 

Or,  since  log,  1  =  0  (Art.  403) , 

log,  2  =  2(.3333333  +  .0123457  +  .0008230  +  .0000653 
+  .0000056  4-  .0000005  +  ••.) 
=  2  X. 3465734  =  .6931468 

=  .693147,  correct  to  the  sixth  place  of  decimals. 
Having  found  loge2,  we  may  calculate  log,  3  by  putting 
m  =  3  and  w  =  2  in  the  result  of  Art.  433. 

Proceeding  in  this  way,  we  shall  find  log,  10  =  2.302585... 

435.  To  calculate  the  common  logarithm  of  a  number^  hav- 
ing given  its  Napierian  logarithm. 

Putting  &  =  10  and  a  =  e  in  the  result  of  Art.  415,  we  have 

logio  m  =  ^^^^  = ^- X  log,  m  =  .4342945  x  log,  m. 

^^^  log,  10      2.302585         ^  ^ 

Thus,  logio  2  =  .4342945  x  .693147  =  .301030. 

436.  The  multiplier  by  which  logarithms  of  any  system 
are  derived  from  Napierian  logarithms,  is  called  the  modulus 
of  that  system. 

Thus,  .4342945  is  the  modulus  of  the  common  system. 

437.  Conversely,  to  find  the  Napierian  logarithm  of  a 
number  when  its  common  logarithm  is  given,  we  may  either 
divide  the  common  logarithm  by  the  modulus  .4342945,  or 
multiply  it  by  2.302585,  the  reciprocal  of  .4342945. 

EXAMPLES. 
Find  the  Napierian  logarithms  of  the  following : 

1.  100.  3.    88.2.  6.    .343. 

2,  ,0001.  1   1325.  6.    .08562. 


354  ALGEBRA. 

XXXVII.    COMPOUND    INTEREST    AND 
ANNUITIES. 

438.  The  principles  of  logarithms  may  be  applied  to  the 
solution  of  problems  in  Compound  Interest. 

Let  P  =  the  principal  in  dollars  ; 
n  =  the  number  of  j^ears  ; 

t  =  the  ratio  to  one  3^ear  of  the  time  during  which  sim- 
ple interest  is  calculated  ;  for  instance,  if  the 
interest  is  compounded  semi-annually,  t  =  ^; 
M  =  the  amount  of  one  dollar  for  the  time  t ; 
A  =  the  amount  of  P  dollars  for  n  years. 

1.    Given  P,  n,  f,  P;  to  find  A. 

Since  the  amount  of  one  dollar  for  the  time  t  is  i2,  the 
amount  of  P  dollars  for  the  same  period  will  be  PR. 

That  is,  the  amount  at  the  end  of  the  1st  interval  is  PR. 
In  like  manner,  the  amount  at  the  end  of  the 

2nd  interval  =  PR  x  R  =  PR"" ; 

3rd  interval  =  PR'  xR  =  PR' ;  etc. 

n 
Since  the  whole  number  of  intervals  is  -■>   the  amount  at 

the  end  of  the  last  one,  in  accordance  with  the  law  observed 

n 

above,  will  be  PR*. 

n 

That  is,  A  =  PR'.  (1) 

By  logarithms,  log^  =  logP-f- logi2.  (2) 

Example.  What  will  be  the  amount  of  $7326  for  3  years 
and  9  months  at  7  per  cent  compound  interest,  the  interest 
being  compounded  quarterly  ? 


COMPOUND  INTEREST  AND  ANNUITIES.       355 

In  this  case, 

P=7326,  n  =  3|,  ^  =  ^,  i2=  1.0175,  and  -=  15. 
logP=  3.8649 
log  i2  =  0.0075  ;  multiply  by  15  =  0.1125 

log^  =  3.9774 
.•.A=  $9492,  Ans, 

2.  Given  n,  t,  jR,  A;  to  find  P. 

From  (2),  log P  =  log ^  -  -  log  i2. 

c 

Example.  What  sum  of  money  will  amount  to  $  1763.50  in 
3  years  at  5  per  cent  compound  interest,  the  interest  being 
compounded  semi-annually  ? 

In  this  case, 

n  =  3,  i  =  |,  P=  1.025,  vl=  1763.5,  and- =  6. 

log^  =  3.2464 
logP  =  0.0107  ;  multiply  by  6  =  0.0642 
logP=  3.1822 
.•.P=  8 1521.40,  Ans. 

3.  Given  P,  ^  i2.  A;  to  find  n. 

From  (2) ,    -  log  P  =  log  ^  -  log  P. 
c 

Whence,  ,,^^0o^-^ -log^). 

\ogR 

Example.  In  how  many  years  will  8  300  amount  to  $  396.90 
at  6  per  cent  compound  interest,  the  interest  being  com- 
pounded quarterly? 

Here,   P=300,  t  =  \,  P=  1.015,  and  ^=396.9. 

.  ^  _  log  396.9  -  log  300  ^  2.5987  -  2.4771  ^  .1216       ' 
4  log  1.015  4  X. 0064       ~  ,0256 

=  4.75  years,  Ans, 


356  ALGEBRA. 

4.    Given  P,  w,  t^  A;  to  find  R. 

From  (2),  log 7?  =  l^ii^liSSf. 

n 

7 

Example.  At  what  rate  per  cent  per  annum  will  $500 
amount  to  $688.83  in  6  years  and  6  months,  the  interest 
being  compounded  semi-annually  ? 

Here,    P=  500,  n  =  ^,  t=^,A  =  688.83,  and  -  =  13. 

log^=        2.8381 

logP=       2.6990 

13)0.1391 

logR=        0.0107 

.'.B=       1.025. 

That  is,  the  interest  on  one  dollar  for  6  months  is  $.025, 
and  the  rate  is  5  per  cent  per  annum. 


EXAMPLES. 

439.  1.  What  will  be  the  amount  of  $1000  for  18  years 
at  6  per  cent  compound  interest,  the  interest  being  com- 
pounded annually? 

2.  What  sum  of  money  will  amount  to  $870.50  in  7  years 
and  3  months  at  3  per  cent  compound  interest,  the  interest 
being  compounded  quarterly  ? 

3.  In  how  many  years  will  $968  amount  to  $1269.40  at 
5  per  cent  compound  interest,  the  interest  being  compounded 
semi-annually  ? 

4.  At  what  rate  per  cent  per  annum  will  $2600  gain 
$416.40  in  3  years  and  9  months,  the  interest  being  com- 
pounded quarterly? 

5.  In  how  many  years  will  a  sum  of  money  double  itself 
at  5  per  cent  compound  iijterest,  the  interest  being  con}' 
pounded  annually? 


COMPOUND  INTEREST  AND  ANNUITIES.       357 

6.  In  how  many  years  will  a  sum  of  money  treble  itself  at 
7  per  cent  compound  interest,  the  interest  being  compounded 
semi-annually? 

7.  What  sum  of  money  will  amount  to  $1000  in  11  years 
and  8  months  at  3J  per  cent  compound  interest,  the  interest 
being  compounded  every  four  months  ? 

ANNUITIES. 

440.  The  present  value  of  a  sum  of  money,  due  at  the  end 
of  a  given  period,  is  the  sum  which  when  put  at  interest  for 
the  period  in  question  will  amount  to  the  given  sum. 

In  finding  the  present  value  of  an  annuity,  it  is  customary 
to  allow  compound  interest. 

441.  To  find  the  present  value  of  an  annuity  to  continue 
for  n  successive  years,  allowing  compound  interest. 

Let   A  =  the  annuity  in  dollars  ; 

li  =  the  amount  of  one  dollar  for  one  year  ; 
P„,  =  the  present  value  of  the  payment  due  at  the  end 

of  m  years ; 
P=  the  present  value  of  the  annuity. 
By  Art.  440,  the  sum  P^  will  amount  to  A  when  put  at 
compound  interest  for  m  years,  the  interest  being  compounded 
annually. 

In  this  case,  w  =  m,  and  t=l  ',  whence  by  (1),  Art.  438, 

A=P^Iir,  or  P^  =  — • 

Br 

By  aid  of  the  above  formula,  the  present  value  of  the 
1st   payment  =  —  ; 

2nd  payment  =  ^ ; 


nth  payment  =  — 


358  ALGEBRA. 

Hence  the  sum  of  the  present  values  of  the  separate  pay- 
ments, or  the  present  value  of  the  annuity,  is 

A        A  iA_j-^ 

Thatis,      P=^[±  +  ^^  +  ...  +  i  +  y. 

The  expression  in  brackets  is  the  sum  of  the  terms  of  a 
Geometrical  Progression,  in  which  a  =  -— ,  r  =  i2,  and  ?  =  7, ; 
whence  by  (II.),  Art.  348, 

Example.  What  is  the  present  value  of  an  annuity  of  $  150 
to  continue  for  20  years,  allowing  4  per  cent  compound 
interest  ? 


Here,  ^=150,  n  =  20,  i2=1.04,  and  i?-l  =  .04. 

mi 

.04  L        (1. 


Whence,  P=l^fi_^^]. 


log L_^  =  20  colog  1.04. 

^(1.04)20  ^ 

colog  1.04  =  9.9830 
20 


9.6600 
Number  corresponding  =    .4571. 

Therefore,  p=  3^  (1  _  .4571)  =  3750  x  .5429. 

log  3750  =  3.5740 
log  .5429  =  9.7347 
log  P=  3.3087 
.-.  P=  $2035.70,  Ans. 


COMPOUND  INTEREST   AND   ANNUITIES.       359 

442.    If  in  (1),  Art.  441,  n  is  indefinitely  increased,  the 
limiting  value  of  the  second  member  is 

A 


E-1 


(Art.  301). 


That  is,  the  present  value  of  a  perpetual  annuity  is  equal  to 
the  amount  of  the  annuity  divided  by  the  interest  on  one  dollar 
for  one  year. 

EXAMPLES. 

443.  1.  What  is  the  present  value  of  an  annuity  of  $  200 
to  continue  15  years,  allowing  5  per  cent  compound  interest? 

2.  What  is  the  present  value  of  a  perpetual  annuity  of 
$600,  allowing  3^  per  cent  compound  interest? 

3.  What  is  the  present  value  of  an  annuity  of  $1127  to 
continue  3  years,  allowing  7  per  cent  compound  interest? 

4.  What  annuity  to  continue  10  years  can  be  purchased 
for  $2038,  allowing  6  per  cent  compound  interest? 

5.  A  person  borrows  $5254;  how  much  must  he  pay  in 
annual  instalments  in  order  that  the  whole  debt  may  be  dis- 
charged in  12  years,  allowing  4^  per  cent  compound  interest  ? 


360  ALGEBRA. 

XXXVIII.    PERMUTATIONS  AND  COMBINA- 
TIONS. 

444.  The  different  orders  iu  which  quantities  can  be 
arranged  are  called  their  Permutations. 

Thus  the  permutations  of  the  quantities  a,  6,  c,  taken  two 

at  a  time,  are  t.         r.     i.  i, 

'  ab,  ac,  ba,  be,  ca,  cb ; 

and  their  permutations  taken  three  at  a  time,  are 
abc,  acb,  bac,  bca,  cab,  cba. 

445.  The  Combinations  of  quantities  are  the  different  col- 
lections which  can  be  formed  with  them,  without  regard  to 
the  order  in  which  they  are  placed. 

Thus  the  combinations  of  the  quantities  a,  6,  c,  taken  two 

at  a  time,  are  ,    , 

'  ab,  be,  ca  ; 

for  though  ab  and  ba  are  different  permutations,  they  form 
the  same  combination. 

446.  To  find  the  number  of  permutations  of  n  quantities 
taken  two  at  a  time. 

Let  the  quantities  be  %,  aa?  <^39  <^45  •'•-)  ^«- 

The  permutations  of  the  quantities  taken  two  at  a  time, 
having  cti  as  the  first  element,  are 

ttitta,  ciiOt.3,  aia^^, ...,  ditt^  j 
the  number  of  which  is  n  —  1. 

In  like  manner,  there  are  n  —  1  permutations  of  the  quan- 
tities takew  two  at  a  time,  having  ag  as  the  first  element ;  and 
similarly  for  each  of  the  remaining  quantities  ag,  a^,  ...,  a„. 

Therefore  the  whole  number  of  permutations  of  the  quan- 
tities taken  two  at  a  time  is  equal  to 

n(w— 1). 


PERMUTATIONS  AND  COMBINATIONS.         361 

447.    We  will  now  consider  the  general  case. 

To  find  the  7iumher  of  permutations  of  n  quantities  taken  r 
at  a  time. 

Let  the  quantities  be 

tti,    0^2'   <^3?    •••>   ^r?   ^r+l1  <^r+2)    •••?   <^n* 

One  of  the  permutations  containing  r  quantities  will  be  that 
consisting  of  the  first  r  quantities  in  their  order  ;  that  is, 

aia.2a^...ar. 

Placing  after  this  the  other  n  —  r  quantities  one  at  a  time, 
as  follows, 


there  are  formed  n  —  r  dififerent  permutations,  each  contain- 
ing r+l  quantities. 

We  may  proceed  in  a  similar  manner  with  the  remaining 
permutations  containing  r  quantities,  and  in  each  case  we 
shall  obtain  n  —  r  permutations  containing  ?-  -f  1  quantities. 

That  is,  the  number  of  permutations  of  the  quantities  taken 
r  at  a  time,  multiplied  by  7i  —  ?*,  is  equal  to  the  number  of 
permutations  of  the  quantities  taken  r  + 1  at  a  time. 

But  the  number  of  permutations  of  the  quantities  taken  two 
at  a  time  is  equal  to  n{n  —  l)  (Art.  446). 

Hence  the  number  of  permutations  of  the  quantities  taken 
three  at  a  time,  is  equal  to  the  number  taken  two  at  a  time, 
multiplied  by  n  —  2,  or  w(n  —  1)  (n  —  2). 

The  number  of  permutations  of  the  quantities  taken  four 
at  a  time,  is  equal  to  the  number  taken  three  at  a  time,  multi- 
plied by  w  —  3,  or  n{n—l) {n  —  2) (/i  —  3) ;  and  so  on. 

We  observe  that  the  last  factor  in  the  number  of  permuta- 
tions is  71,  minus  a  number  one  less  than  the  number  of  quan- 
tities taken  at  a  time. 


362  ALGEBRA. 

Hence  the  number  of  permutations  of  the  quantities  taken 
r  at  a  time  is  given  by  the  formula 

w(n-l)(n-2)...[n-(r-l)], 
or,  ?i(n-l)(n-2)...(n-r  +  l).  ^  (1) 

448.  If  all  the  quantities  are  taken  together,  r  =  n,  and 
formula  (1)  becomes 

n{n-l){n-2)'"l^\n.  (2) 

That  is,  the  number  of  permutations  of  n  quantities  taken 
n  at  a  time  is  equal  to  the  product  of  the  natural  numbers  from 
1  to  n  inclusive.     (See  Note  2,  Art.  363.) 

449.  To  find  the  number  of  combinations  of  n  quantities 
taken  r  at  a  time. 

The  number  of  permutations  of  n  quantities  taken  r  at  a 
time  is 

n(n-l)(n-2)...(7i-r+l)  (Art.  447). 

But  by  Art.  448,  each  combination  of  r  quantities  may 
have  [r  permutations. 

Hence  the  number  of  combinations  of  n  quantities  taken  r  at 
a  time  is  equal  to  the  number  of  permutations,  divided  by  \rj, 
that  is, 

n(n-l)(n-2)...(7i-r  +  l)  /g) 

ll 

450.  Multiplying  both  terms  of  (3)  by  the  product  of  the 
natural  numbers  from  1  to  n  —  r  inclusive,  we  have 

yi(?i-l)...(?i-r+l)X(n-r)---3-2»l^       [^j: 

[r  X  1  •  2 . 3  •  •  •  (n  —  ?')  \r_ \n  —  r ' 

which  is  another  form  of  the  result. 

451.  By  Art.  450,  the  number  of  combinations  of  n 
quantities  taken  ?i  —  r  at  a  time,  is 

In  \n 


\n  —  r\n  —  {n  —  r)          \n  —  r\r 


PERMUTATIOKS  AND  COMBINATIOJTS.         363 

But  this  is  the  same  as  the  number  of  combinations  of  n 
quantities  taken  r  at  a  time  (Art.  450) . 

Hence,  the  number  of  combinations  of  n  quantities  taken  r 
at  a  time  is  equal  to  the  number  of  combinations  of  n  quanti- 
ties taken  n  —  r  at  a  time.- 

EXAMPLES. 

452.  1.  How  many  changes  can  be  rung  with  ten  bells, 
taking  7  at  a  time  ? 

Here  n  =  10,  r  =  7,  and  ?i  —  ?•  4- 1  =  4. 
Then  by  (1),  Art.  447,  the  required  number 
=  10. 9. 8.7.6. 5.4  =  004800,  Ans. 

2.    How  many  different  combinations  can  be  formed  with 

16  letters,  taking  12  at  a  time? 

By  Art.  451,  the  number  of  combinations  of  16  quantities 
taken  12  at  a  time  is  equal  to  the  number  of  combinations 
of  16  quantities  taken  4  at  a  time. 

Putting  n  —  16  and  ?'  =  4,  in  (3),  Art.  449,  we  have 
16.15.14.13 


1.2.3.4 


1820,  Ans, 


3.  How   many   permutations    can    be   formed   of   the    26 
letters  of  the  alphabet,  taken  5  at  a  time  ? 

4.  How  many  permutations  can  be  formed  of  the  letters 
in  the  word  forming^  taken  all  together  ? 

5.  How    many    combinations    can  be    formed    with    the 
letters  in  the  word  triayigles,  taking  four  at  a  time? 

6.  How  many  different  numbers,  of  five  different  figures 
each,  can  be  formed  with  the  digits  1,  2,  3,  4,  5,  6,  7,  8,  9? 

7.  From  a  company  of  40  soldiers,  how  many  different 
pickets  of  6  men  can  be  taken? 


364  ALGEBRA. 

8.  How  many  combinations  can  be  formed  with  18  quan- 
tities, taking  11  at  a  time? 

9.  How  many  different  words  of  4  letters  each  can  be 
made  with  6  letters?  How  many  of  3  letters  each?  How 
many  of  6  letters  each  ?     How  many  in  all  possible  ways  ? 

10.  How  many  combinations  can  be  formed  with  24  letters, 
taking  18  at  a  time? 

11.  How  many  different  committees,  consisting  of  8  per- 
sons each,  can  be  formed  out  of  a  corporation  of  20  persons? 

12.  How  many  different  numbers,  of  4  different  figures 
each,  can  be  formed  from  the  digits  1,  2,  3,  4,  5,  6,  7,  8, 
9,  0? 

13.  How  many  different  words,  each  consisting  of  4  con- 
sonants and  2  vowels,  can  be  formed  from  8  consonants  and 
4  vowels? 

The  number  of  combinations  of  the  8  consonants,  taken  4 
at  a  time,  is  o   r^   n    n 

1.2.3.4 
The  number  of  combinations  of  the  4  vowels,  taken  2  at  a 
time,  is  .0 

But  any  one  of  the  70  sets  of  consonants  may  be  associated 
with  any  one  of  the  6  sets  of  vowels. 

Hence  there  are  in  all  70  x6,  or  420  sets,  each  containing 
4  consonants  and  2  vowels. 

Now  each  of  these  sets  of  6  letters  may  have  |^,  or  720, 
different  permutations  (Art.  448). 

Therefore  the  whole  number  of  different  words  is 
420  X  720  =  302400,  Ans. 

14.  How  many  different  words,  each  consisting  of  3  con- 
sonants and  1  vowel,  can  be  formed  from  12  consonants  and 
3  vowels? 


PERMUTATIONS  AND  COMBINATIONS.         365 

15.  How  many  different  committees,  each  consisting  of  2 
Republicans  and  3  Democrats,  can  be  formed  from  14  Repub- 
licans and  21  Democrats? 

16.  Out  of  9  red  balls,  4  white  balls,  and  6  black  balls, 
how  many  different  combinations  can  be  formed,  each  con- 
sisting of  5  red  balls,  1  white  ball,  and  3  black  balls? 

17.  How  many  different  words,  each  consisting  of  4  con. 
sonants  and  3  vowels,  can  be  formed  from  10  consonants, 
and  5  vowels? 

18.  Out  of  11  physicians,  13  teachers,  and  8  lawyers, 
how  many  different  committees  can  be  formed,  each  consist 
ing  of  3  physicians,  4  teachers,  and  2  lawyers? 

19.  How  many  words  of  seven  letters  each  can  be  formeu 
from  the  letters  a,  Z>,  c,  d,  e, /,  g^  each  word  being  such  that 
the  letters  a,  6,  c  are  never  separated  ? 


366  ALGEBRA. 

XXXIX.    CONTINUED    FRACTIONS. 
453.    A  continued  fraction  is  an  expression  of  the  form 

a-\ — -; 

d 

e  +  ... 
or,  as  it  is  usually  written  in  practice, 

h      d 


We  shall  limit  ourselves  in  the  present  chapter  to  con- 
tinued fractions  of  the  form 

,     1       1 

5+C+... 

where  each  numerator  is  unity,   a  is  0  or  any  positive  integer, 
and  each  of  the  quantities  6,  c,  ...,  is  a  positive  integer. 

454.  A  terminating  continued  fraction  is  one  in  which  the 
number  of  denominators  is  finite  ;  as, 

,111 

a-\ -• 

h-\.  c+  d 

It  may  be  reduced  to  an  ordinary  fraction  by  the  process 
of  Art.  161. 

An  infinite  continued  fraction  is  one  in  which  the  number 
of  denominators  is  indefinitely  great. 

455.  In  the  continued  fraction 
1      1        1 


«!  + 


^2+  «3+  a4+- 
a,  is  called  the  first  convergent ; 

ttiH —  is  called  the  second  convergent; 

«2 

tti  + is  called  the  third  convergent;  and  so  on. 

^2+   «3 


CONTINUED   FRACTIONS.  367 

Note.   If  ai  =  0,  as  in  the  continued  fraction 
1      1         1 

y 

«3  +  «3  +  tti  +"• 

then  0  is  considered  the  first  convergent. 

456.   Any  ordinary  fraction  in  its  lowest  terms  may  he  con- 
verted into  a  terminating  continued  fraction. 

Let  the  given  fraction  be  -• 

Divide  a  by  6,  and  let  aj  denote  the  quotient  and  bi  the 
remainder ;  then, 

a  ,  bi  ,1 

_,,  +  ^  =  a,  +  ^. 

Divide  b  by  5i,  and  let  ag  denote  the  quotient  and  62  tl^e 
remainder ;  then, 

?  =  a.  +  -l-  =  a.  +  -!— . 

bi  Oi 

Again,  divide  61  by  62?  and  let  as  denote  the  quotient  and 
63  the  remainder  ;  then, 

?  =  a,  + i— -  =  «,+  ' 


b  ,       1  ^  ,       1 


O2  ©2 


The  process  is  the  same  as  that  of  finding  the  Highest 
Common  Factor  of  a  and  b  (Art.  129);  and  since  a  and  b 
are  prime  to  each  other,  we  must  eventually  obtain  a  remain- 
der unity,  at  which  point  the  operation  terminates. 

Hence  any  ordinary  fraction  in  its  lowest  terms  can  be  con- 
verted into  a,  terminating  continued  fraction. 


368        •  ALGEBRA. 

62 
Example.    Convert  —  into  a  continued  fraction. 
Zo 

23)62(2  =  ai 
46 

16)23(1  =  a2 
16 

T)l6(2  =  a3 
U 

6 

1 

Therefore,     — =2+— -^i^,   Ans. 

'     23  1+2+3+2 

457.    A  quadratic  surd  (Art.  250)  may  he  converted  intc 
an  injinite  continued  fraction. 

Example.    Convert  ■^y6  into  a  continued  fraction. 

The  greatest  integer  in  ^6  is  2  ;  we  then  write 

V6=:2+-(V6-2). 

Reducing  -y/Q  —  2  to  an  equivalent  fraction  with  a  rational 
numerator  (Art.  244) ,  we  have 

^  ^  V6  +  2  V6  +  2 

=  2 +--7^ (1) 

V6  +  2 

2 
The  greatest  integer  in ^^  is  2  ;  we  then  write 

V6+-2^^  I  V6-2 
2  2 

^g      (V6-2)(V6  +  2)^^  1 

2(V6+-2)  V6  +  2 


CONTINUED  FRACTIONS.  369 

Substituting  in  (1), 

1_ 

1 


V6  =  24- —, (2) 

2  + 


V6  +  2 

The  greatest  integer  in  ^6  +  2  is  4  ;  we  then  write 
V6  +  2  =  4  +  ( V6  -  2)  =  4  +  i^«^lMV6  +  2i 

-y/6  4"  2 


V6+2  V6+2 


Substituting  in  (2),  we  have 

1 


V6  =  24- 


2  + ' 


44         ' 


V6  +  2 
2 

The  steps  now  recur,  and  we  have 

V6  =  2-f— —  J^ —^   Ans. 

^  2+4+2+4+- 

Note.   An  infinite  continued  fraction  in  which  the  elements  recur,  is 
called  a  periodic  continued  fraction. 

458.   A  periodic  continued  fraction  may  always  he  expressed 
as  the  root  of  a  certain  quadratic  equation. 

Example.  Express as  the  root  of  a  certain 

^         1+3+1+3+" 

quadratic  equation. 

Let  X  denote  the  value  of  the  fraction  ;  then, 

^  ^  J 1_  ^     3+- a;     ^  3  +  a? 

1+3H-X     3  +  a;  +  l      4  + a;* 

Clearing  of  fractions, 

^x  +  x^=^^+x,  or  ar'-f  3rc  =  3. 


370  ALGEBRA. 

Solving  the  equation, 


3+V9  +  12_-3+V21^    ^ns. 


2  2 

Note.  The  +  sign  is  taken  before  the  radical,  since  x  is  evidently  a 
positive  quantity. 

PROPERTIES    OF    CONVERGENTS. 

459.  Let  the  continued  fraction  be 

,11  11 

and  let  p^  denote  the  numerator,  and  q^  the  denominator,  of 
the  rth  convergent  (Art.  455)  when  expressed  in  its  simplest 
form. 

460.  To  determine  the  law  of  formation  of  the  successive 
conver gents. 

The  first  convergent  is  dp 

The  second  is  a^-\--  =  Mdti. 

The  third  is     a^^—l-  =  a,  +  — 'h_ ^M^^±ai±^. 

«2+  %  «2«3  +  1  <*2<^3  +  1 

The  third  convergent  may  be  written  in  the  form 
(aia2  +  l)a3  +  <^. 

a2«3  H-  1 
in  which  we  observe  that 

1.  The  numerator  is  equal  to  the  numerator  of  the  preced- 
ing convergent^  multiplied  by  the  last  denominator  taken,  plus 
the  numerator  of  the  convergent  next  but  one  preceding. 

2.  The  denominator  is  equal  to  the  deiwminator  of  the 
preceding  convergent,  multiplied  by  the  last  denomiriator  taken, 
plus  the  denominator  of  the  convergent  next  but  one  2^TQceding. 

We  will  now  prove  by  Induction  (Note  1,  Art.  363)  that 
the  above  laws  hold  for  all  convergents  after  the  second, 
when  expressed  in  their  simplest  forms. 


CONTINUED  FRACTIONS.  371 

Assume  that  the  laws  hold  for  all  convergente  as  far  as 
the  nth. 
The  nth  convergent  is 

Then  since  the  last  denominator  is  a„,  we  have 

Pn  =  (InPn-l  +Pn-2  »    and    g„  =  tt^^^.i  +  qn-2'  (1) 

Whence,  P_,^<^nP.-.+P.-.,  (2) 

Qn  anqn-l  +  qn-2 

The  (n  4-l)st  convergent  is 

J 1_        1      1 

^     a2+a3+      a„+«Xn+i' 

which   differs   from  the   nth   only   in   having   a^^H ,  or 

^^        ,  m  place  of  a„. 

Substituting  ^"^"+'  "*"  ^  for  a,  in  (2),  wehave 

;; Pn-1  +  i5n-2 


?n-l  -r  yn-2 

_  On+1  {anPn-l-\-Pn-2)  +  Pn-1 

It  is  evident  that  the  second  member  of  (3)  is  the  simplest 
form  of  the  (n  +l)8t  convergent,  and  therefore 

i>n+l  =  «n+lPn+i>n-l,    aud    Qn+l  =  a^^+lQu -^  Qn-l' 

These  results  are  in  accordance  with  the  laws  stated  on  the 
preceding  page. 

Hence,  if  the  laws  hold  for  all  convergents  as  far  as  the 
nth,  they  also  hold  as  far  as  the  (n  -f  l)st. 


372  ALGEBRA. 

But  we  know  that  they  hold  as  far  as  the  third  convergent, 
and  hence  the}^  also  hold  as  far  as  the  fourth ;  and  since  they 
hold  as  far  as  the  fourth,  they  also  hold  as  far  as  the  fifth ; 
and  so  on. 

Therefore  the  laws  hold  for  all  convergents  after  the  second. 

Example.    Find  the  first  five  convergents  of 

1+  2+  3+  4+- 

The  first  convergent  is  1,  and  the  second  is  1  +  1,  or  2. 

Then  by  aid  of  the  laws  just  proved, 

^,     ^i-   -,  .      2-2+1         5 

the  tlurd  is    ' —    =  -  ; 

1-2+1         3 

^v,    *      ^1.-    5-3  +  2        17 
the  fourth  is ' —    =  —  ; 

3.3+1       10 

the  fifth  is     !—  =  — 

10-443      43 

461.  The  difference  between  two  coyisecutive  convergents 
—  and  -^^  is  equal  to 

qn  Qn+l  Qnqn+l 

The  difference  between  the  first  and  second  convergents  is 

/  1\  1 

ai  H —   —  tti  =  — 

Thus  the  theorem  holds  for  the  first  and  second  conver- 
gents. 

Assume  that  it  holds  for  the  nth  and  (n  +l)st  convergents  ; 
that  is, 

Then, 

Pn+l  _^Pn+2  _Pw+l   _^  f^n+2Pn+\  4"  j^n    /^j.^     460) 
9'w+l         9'n+2         9'n+l         C^^+2  5'n+l  +  Q'n 


CONTINUED  FRACTIONS.  373 

^    ((^n+2Pn+lQn+l-{-Pn+iqn)  ^  (cin+2Pn+iqn+l  +i>»gn+l) 
Qn+l{a^+2gn+l+qn) 

^Pn^.qn-^Pnqn^,  (Art.  460)  =3— i—,    by  (1). 

Q'n+1  qn+2  q,i+l  qn+2 

Hence  if  the  theorem  holds  for  any  pair  of  consecutive  con- 
vergents,  it  also  holds  for  the  next  pair. 

But  we  know  that  it  holds  for  the  first  and  second  conver- 
gents,  and  hence  it  also  holds  for  the  second  and  third ;  and 
since  it  holds  for  the  second  and  third,  it  also  holds  for  the 
third  and  fourth  ;  and  so  on. 

Therefore  the  theorem  holds  universall}-. 

462.  It  follows  from  Art.  461  that  p„  and  q^  can  have  no 
common  divisor  except  unity ;  for  if  they  had,  it  would  be  a 
divisor  of  p„</„+i~i5n+i7n?  or  unity,  which  is  impossible. 

Therefore  all  convergents  formed  in  accordance  with  the 
laws  of  Art.  460  are  in  their  lowest  terms. 

463.  The  even  convergents  are  greater^  and  the  odd  con- 
vergents less^  than  the  fraction  itself. 

I.  The  first  convergent,  a^,  is  les^  than  the  fraction  itself, 

since is  omitted. 

a2  +  ... 

II.  The  second,   aiH — ,  is  orrea^er,  because  its  denomina- 

tor  as  is  less  than  a^,  H ,  the  denominator  of  the  fraction. 

a3+... 

III.  The  third,  aiH ,  is  less,  because,  by  II.,  the  de- 

nominator  a^-\ —  is  greater  than  a^ H ,  the  denom- 

ttg  a^^a^^... 

inator  of  the  fraction  ;  and  so  on. 

Hence  the  first,  third,  ...,  convergents  are  less,  and  the 
second,  fourth,  ...,  convergents  greater  than  the  fraction 
itself. 


874 


ALGEBRA. 


464.    Any  coyivergent  is  nearer   than   the  preceding   con- 
vergent to  the  value  of  the  fraction  itself. 

By  Art.  460,  ^^^  =  «>.+2Pn4-i  +Pn 


Qn+2         <^n+1  ^n+1  +  ^n 


The  fraction  itself  is  obtained   from  its   (n-j-2)nd   con- 
vergent by  putting  a^+o  -\ in  place  of  a„^9. 

Hence,  denoting  the  value  of  the  fraction  itself  by  x^  we 

have 

1 

Pn+l+Pr. 

mPn+i-hPn 


an4-2  4- 


«n+34- 


hi+2 


+ 


1 


where  m  stands  for  a„^2  + 


1 


'tnqn+i  +  Qr, 


«n+3  +  - 


Now,  ^^Pn^^i>n-H+P.^P, 

Wl(Pn4-ign~P»gn+l) 


qn{mqn+i 
m 


Also,    a; 


qn+1         mq^+l  +  gn  '^  ^n+l 


qn) 
(Art.  461). 


(1) 


(2) 


qn+i  i^nq^+i  +  qn)      qn+1  i'inqn+i  +  qn) 

Since  <x„+2  is  a  positive  integer,  a^^g  H is  >  1  ;  that 

is,  m  is  >  1. 

And  since  g„+i  =  a^+^q^  +  g„_i  (Art.  460) ,  q^+i  is  >  g„. 

Therefore  the  fraction  (2)  is  less  than  the  fraction  (1),  for 
it  has  a  smaller  numerator  and  a  greater  denominator. 

Hence  the  (n+l)st  convergent  is  nearer  than  the  nth  to 
the  value  of  the  fraction  itself. 


CONTINUED   FRACTIONS.  375 

465.    By  Art.   464,  the   difference  between  the  fraction 
itself  and  its  nth  convergent  is 

— ; ^,  or  — -.  (1) 

Since  m  is  >  1  (Art.  464),  the  denominator  qn(Qn+i  + 

The  denominator  is  also  >  q^qn+i- 

Hence  the  fraction  (1)  is  > — ; r^?  and    < 

Qn{qn+,  +  qn)  qnqn+l 

That  is,  the  eiror  made  in  taking  the  nth  convergent  for  the 
fraction  itself  lies  between  the  limits 

1  ,       1 

and 


^■Mn+l-hqn)  qnqn+l 

EXAMPLES. 

466.  Convert  each  of  the  following  into  a  continued  frac- 
tion, and  find  in  each  case  the  first  five  convergents : 

1.   ^.  3.    3.61.  5.    ^.  7.   if«. 

39  326  345 

2     72  4     112  6    —  8     ?^. 

91*  '    153*  '     89  *  *    6961* 

Convert  each  of  the  following  into  a  continued  fraction, 
find  in  each  case  the  first  four  convergents,  and  determine 
limits  to  the  error  made  in  taking  the  fourth  convergent  for 
the  fraction  itself : 

9.  V5-  10.  V3.  11-  Vll-  12.  V7. 
Express  each  of  the  following  in  the  form  of  a  surd : 
i«      1      1      1        1  15     24-- —' 

^^-  ^3T^3T:^*  1^-  ^  +  1+1  +  ... 

^       1+4+1+4+-  ^1+8+  1+8+- 


376  ALGEBRA. 

17.  The  ratio  of  the  circumference  of  a  circle  to  its  diame- 
ter is  approximately  equal  to  3.14159  ;  express  this  decimal 
as  a  continued  fraction,  and  find  the  first  four  convergents. 

18.  The  modulus  of  the  common  system  of  logarithms  is 
approximately  equal  to  .43429  ;  express  this  decimal  as  a 
continued  fraction,  find  its  seventh  convergent,  and  determine 
limits  to  the  error  made  in  taking  this  convergent  for  the 
fraction  itself. 

19.  The  base  of  the  Napierian  system  of  logarithms  is 
2.7183  approximately;  express  this  decimal  as  a  continued 
fraction,  find  its  eighth  convergent,  and  determine  limits  to 
the  error  made  in  taking  this  convergent  for  the  fraction 
itself. 

20.  Express  the  positive  root  of  the  equation 

a^-a;-ll  =  0 
as  a  continued  fraction,  and  find  the  first  five  convergents. 


ANSWERS. 


Note.   In  the  following  collection  of  answers,  all  those  are  omitted 
which,  if  given,  would  destroy  the  utility  of  the  example. 


1. 

19. 

2. 

1. 

3. 

24. 

4. 

15f. 

6. 

iZ. 

12  144 


Art.  40  ;  pages  8  and  9. 

6.  — .  11.  48. 
16 

7.  6|.  12.  114. 

8.  9.  13.  24. 

9.  5^.  14.  204. 
135 

10.    IM?.  15.  310.                20.    48. 


16. 

4. 

17. 

9^ 
5' 

18. 

50 
39' 

19. 

36. 

21.   3.  22.    ^' 


Art.  43  ;   page  12. 

4.  35  and  7.  8.    A,  $20;  B,  $60;   C,  $180. 

5.  A,  31  ;  B,  37.  9.    52  and  73. 

6.  A,  $536;  B,  $664.  10.    13,  39,  and  46. 

7.  $0.93.  11.    A,  $28;  B,  $43  ;  C,  $56. 

12.    Horse,  $275;  carriage,  $100;  harness,  $25. 
13.    15,  45,  and  48.  14.    35,  70,  and  105. 

15.    Cow,  $45;  sheep,  $18;  hog,  $12. 

Art.  57  ;   page  19. 
4.    2a-26-f2d.  6.    4a^  7.    2a^-ab. 


2  ALGEBRA. 

8.  5a^-Sx^-e.  12.    -ar^  +  3a;+2. 

9.  6mn-ab-4:C—x+Sm\     13.    3a-j-Sb  +  3c  +  ^d. 

10.  x-6y,  14.    5a3-3(x62_453^ 

11.  _a;  +  3m-7n.  15.    a3-4a^. 

Art.  64 ;  pages  22  and  23. 

6.  4a5.  10.  66  +  1. 

7.  _4a6c-14a;-22/-148.  11.  4m-8w-r  +  3s. 

8.  2m-42/2  +  12a  +  l.  12.  6d-26-3a-3c. 

9.  14a^-8^2^5a5-7.  13.  57n^  +  9n^ -71x. 

14.    _3a-26  +  10c-13d  +  2a;. 

15.  2a-6-3c.  20.    2a;-3^. 

16.  x^-a^-Say'-Sx+n.     21.    ea^ +  9a^ -^a^ -a-6. 

17.  4a3-6a26-2a62-96^    22.    -3a^-5i»V+8a;2/2+22/3. 

18.  4a^-3a3+6a2-6a+3.     23.    5a^  + 7a^  +  18.T.. 

19.  ^x^-xy.  24.    -a2-4a6  +  6^ 

25.  5a^  +  6iC2/-7/-6a;— 72/  +  6. 

26.  SxP-6x^-a^-na^  +  Ax-2. 

27.   4ajs+ 90^2/ -4a;/- 32/3.     28.    4a^  + 7a3_2a-4. 

Art.  69 ;  pages  25  and  26. 

4.  5x-y.  10.   a+b-c-{-d-e.  16.  -3a-l. 

5.  a-b  +  c.  11.   2/.  17.  4a;-2. 

6.  om2-6w-4a.  12.    14icH-2.  18.  67/1  +  2. 

7.  _a2-362.  13.    6m-3ri.  19.  0. 

8.  2a +  2.  14.    2a; +  42/.  20.  a +  26. 

9.  a— 6+c— c?+e.  15.    a  —  c. 

Art.  82  ;   pages  32  to  34. 

4.  16x^-Ux-U.  6.    -6a2+i6a6-862. 

5.  -12a;2_f_28a;-15.  7.    50a;22^2_i8^ 


ANSWERS. 


8.  W-a\  11.   6a:«-16a^2/+6a;/4-4/. 

9.  10a*62_3^353_i8^2j4^      12.    2m*- 8m^7i  +  18m«^ 
10.    ax^  —  a.  13.    iK*  +  4aj4-3. 

14.    30a-^-43a26  4-39a52-206^ 
15.    8a^-14a^-18a;  +  21.      16.    a^ -62^_26c-c2. 

17.  2iB*-a^  +  8a;-5. 

18.  6a^  +  13a^-70a^  +  71a;-20. 

19.    6fl^-19a^  +  22a;H-5.       20.    -m«-37m2+70m-50. 

21.  -6ar'-25a^4-7ar'^H-81a^4-3a;-28. 

22.  2  a''^^  -  3  a*63  _  7 ^354  ^  4  ^255^ 

23.  4a^'"+V-16a;'"+<5?/''+^  +  12ar'/»*-^ 

24.  a;*  +  a.V  +  2/'.  26.    12a^+7a^+5cc'+10a;-4. 

25.  16a*4-4a262  +  6^  27.    ?7i6 - 3 m^n  +  mn«- 3 n«. 

28.  243ar^-81a^2/-3a?2/^  +  2/^- 

29.  a** -ba^b  +  lOa^V" -  lOa-ft^  +  5a6*  -  6^ 

30.  a^  +  2/3^^_3^y2^ 

31.  6a^--lla^+14a;*-12a^  +  6a:2-23a;  +  5. 

32.  a^ft^  +  c^d^-a^c^-ft^c^.  36.  36m3-49m  +  20. 

33.  8a«-98a*+152a2-32.  37.  9fl;*-13a^  +  4. 

34.  ic3-6x2-19a;+84.  38.  a^  +  aj^  +  l. 

35.  a«-6«.  39.  a8-6«. 

40.  ??i*-5m2  +  4. 

41.  120x*  +  26a^-lllaj2-14a;H-24. 

42.  a«-6a*62  +  9a25*-46«. 


Art.  83 ;  pages  34  and  35. 

2.  a^H-  ^2  4.  c2  +  d^  +  2a6  +  2ac  4-  2ad  +  26c  +  2hd  +  2cd. 

3.  d&H-ac  — 2acZ  — 26c  +  &d+cd.  4.    a^. 

5.  4a&+46c.     7.    a^  -  2  a'^h'' +  h\     9.    4.a-h2a? +  2aK 

6.  6^  8.    1-a^.  10.    a;2-4a;2/+42/*^-92;2. 


4 

ALGEBRA. 

11. 

a^-y\ 

13.  4a2+462+4c2. 

15. 

-4  2/2;, 

12. 

b'-dK 

14.  ab-{-bc-\-ca-a^-b^-c\ 

16. 

0. 

17. 

Sac. 

18.    -9m*  +  82mV-9n*. 
Art.  93  ;   pages  41  to  43. 

19. 

0. 

4.    2ic-5.  7.    8-5a;.  10.    3a;  — 1. 

6.    2  +  aa;.  8.   Sb^-Aa"".  11.    4m2-10m  +  7. 

6.    a -2b.  9.    -2a;2_2aa;.     12.    9a^-3a;y+2/^ 

13.  8m3  +  4m2+277i  +  l.        15.    4a2  4- 12a6  +  962. 

14.  a  —  b~c.  16.   x^  —  xy  +  ff. 

17.  a;-3.  19.    2«3-3a;-6.     21.    2a  +  3. 

18.  Am^-^n\         20.    2a^-7a;-8.    22.    a^-3a;-2/. 
23.    a6-3a^6  +  9a262_276^   24.   a;-2/  +  0. 

25.  3a^+6a^-2a;-4. 

26.  yix'-a^y^x'y'-xy^  +  y'). 
27.    5m2-4m  +  3.  28.    ^x^-2x-{-l. 

29.    3(a4  4-a^&  +  a262  +  a?>3_f.j4)^ 

30.  8a^4-4a;+l.      33.  a^-2a;-l.       36.  2a^-a;+l. 

31.  9a^-9a;-4.     34.  3a^-5a??/-2r.  37.  3a;2  + 6ic  +  9. 

32.  a^-2a;2/+/-     35.  a2_|_3a64-5  52.    38.  a:2_2^_3. 

39.    m^-m^-Um +  24. 

40.  a; +  2/.  50.   x-\-a. 

41.  o?y  —  xy'^.  51.    (5+c)a  +  6c. 

42.  ar^-2i»2  +  a;-2.  52.    (0^  +  2/) -3. 

43.  o?-2a%-bab^  +  lW.     53.    (a  +  &)2- (a +  ?>)  + 1. 

44.  a^-2i»2_^_|_i^  54^    ^_^^^ 

45.  a;  +  22/-3^.  55.  (m-n)2+2(m-n) +1. 

46.  a'*-^>"*  +  c^  56.  a;2_|.  (^  _5)a;- a6. 

47.  a^  +  2a^-a;  +  l.  57.  a^-6a;4-c. 

48.  2o?-ab  +  2b\  58.  a(^>+c)-6c. 


ANSWERS. 

Art.  96  ;  page  46. 

30.  aP-y^  +  2xz-i-z\  34.    a^ -b^ -2ac  +  (^, 

31.  x'-y^-2yz-z\  35.    a^-2a'  +  l. 

32.  l-a^  +  2ab-b\  36.   x^- 4:X^-\-12x-9. 

33.  x^-a^-2x-l.  37.   m^  +  m'n^  +  n\ 

Art.  Ill ;  pages  56  and  57. 

26.  (x-\-y  +  2){x-\-y-2).    29.    (a  +  64- c)  (a-6 -c). 

27.  (a-6  +  c)(a-6-c).     30.    (c  +  cZ4- l)(c  +  d- 1). 

28.  (a  +  6-c)(a-6  +  c).     31.    (S +  x-y){S -x  +  y). 

32.  (2m2  +  26-l)(2m2-26  +  l). 

33.  (2a-6  +  3(Z)(2a-6-3d). 

34.  {a-{-b  —  m  +  n)(a  —  b  —  m  —  n), 

35.  (a;  +  ?/  — c  — d)  (a;— 2/  — c  +  d). 

36.  {a  +  b-\-m  —  n){a  —  b-{-m-\-n), 

37.  (a  +  6H-c-f  d)(a  — ?>  +  c  — d). 

Art.  120  ;  page  64. 
27.    2xy(x  +  y-{-z){x-{-y-z). 
40.    (x-\-y-j-z){x-\-y-z){x-y-\-z)(x-y^z). 
46.    (a  +  6)(a_&)(x-f2/)(aj-2/). 

50.  {x-2){x-\-S){x'-x-\-6). 

51.  (a-l)(a-2)(a  +  4)(a  +  5). 

53.    (a  +  6  -f  c)  (a  +  6  —  c)  (a  —  6  +  c)  (a  —  6  —  c) . 

55.  {x-iy{x-{-l).  58.  (x  +  2)2(a;-2)2. 

56.  2{a-b)(a  +  2b).  59.  {x-yy. 

60.  (a -l)(a  +  2)(a-2)(a  +  3). 

Art.  131 ;  pages  73  and  74. 

1.  x-2.  4.  8a;-7a.        7.  5m  +  3.  10.  2a;-l. 

2.  2x-i-3.        5.  2a-5.  8.  2a-3a;.  11.  2m-5n. 

3.  a-1.  6.  x^-mx.         9.  a;-2.  12.  a;  +  2. 


e  ALGEBRA. 

13.  ax^-ax.       15.  a^-a-1.     17.  3a; +  2.      19.  x  +  l. 

14.  x^-hx+1.     16.  a^-2a;.         18.  a-x.        20.  a;-2y. 

21.  2a;-3.  22.  3a  +  6. 

Art.  132 ;  page  74. 

1.    2a;4-7.         2.    2a;-5.         3.    3m  +  2ti.         4.    3a-l, 
5.   aj  +  4.  6.   a;-l.  7.   2a  +  3. 

Art.  136 ;  page  76. 

4.    270mV.  7.    336a^2/'2?.  9.    iSOm^n'a^y^ 

6.    210 a6c.  8.    360 aVd^.  10.    252  a^2/'^^ 

11.    lOSOa^ftVd*. 

Art.  137 ;  pages  76  and  77. 

2.  y(x^-y^)'  9.  {x  +  3a)(x-5a)(x-\-7a). 

3.  (aj2_i)(a;_8).  10.  ^(m^-n^). 

4.  24:ab{a'-b^).  11.  (a  + 6)  (a -36)  (a; -2). 

5.  (m -\- n)  (m^  —  n^) .  12.  aa?(a;4- a)  (o^  — a^). 

6.  a2-4a6  +  36^  13.  2^a  +  by{a-by. 

7.  a^(a;  +  2/)(x-?/)2.  14.  ax(^x-S)(x-7){x-\-8). 

8.  12a6c(a2-62).  15.  x4_2a;2  4.i. 

16.  24(1 -a;^). 

17.  (a;  +  l)(a;-2)(a;+3)(aj4-4). 

19.  (2m  +  l)(2m-l)2(477i2_^2m  +  l). 

20.  a\a-l){a'-^l). 

21.  (a_l)(a-3)(a  +  4)(a-5). 

22.  (l+a;)2(l -0^)2(1+0^)2. 

23.  (a  +  &  +  c)(a  +  & -c)(a-6-c). 

24.  Sab(a-b)(x-yy. 

25.  2aa;2(3a;  +  2)2(9ar^-6a;  +  4). 

26.  (ic  +  2/  +  ^)(a^4-2/-2;)(a^'-2/+^)- 


ANSWERS.     .  7 

Art.  139 ;  page  79. 

2.  4a^-13a;  +  6.  4.    24ic3+26a^- 219cc- 56. 

3.  24a;3+22aj2-177a;+140.     5.    2aa;(6a^+a^-42ic-45). 

6.  a*- 16a36  +  86a262_  176^^3 _^1056^ 

7.  27i(2m^-5??i^  +  3m3-5?7i2  +  4^_^4). 

8.  a(30a^-llaa^-59a2a;+12a3). 

9.  2a^  +  a3-17a2-4a+6. 

10.    2aj=+3a;*-4ar5+5a;-6.       11.    a^  -  a362 _(- a^ft^ _  2,5^ 

12.  aa;(ar^  +  a;^-a^-3a^-3aj-l). 

13.  a;(6ar^-31a;^-4ic3  4-44a;2_,_7^_lQ)^ 

14.  ic«+2a;5-4«^-7a^-16a:2_|_32a._8. 

Art.  140 ;  page  79. 

1.  4a^-4a^-39ar^  +  4a;  +  35. 

2.  18a*-33a3+14a2  +  3a-2. 

3.  20a.'*-24a^-51a^  +  41ic-6. 

4.  2a:2^i2ic^-32a;»-29ar'  +  57a;-18). 

5.  a^  +  3a*-23a3-27a2+166a-120. 


12. 
13. 
14. 
15. 
16. 
17. 


Art.  149 ;  pages  83  and  84. 

cd 
3xy 

18. 

m  — 2 
m  +  9 

24. 

9/4-152/+25 
3?/ -5 

7? 
2/ 

19. 

a(37i-f2) 
6(3n-2) 

25. 

a;-l 

2cc+l 

2^y 

20. 

a  +  2& 
a  +  36 

26. 

1 

x-h 

aa;(a;  +  4) 

a-5 

a  +  7 

21. 

2a;H-2/. 

27. 

a  +  64-c 
a  —  ?>  +  c 

2a2^a6 
a6  +  262 

22. 

a^  +  a6  +  62 

28. 

1. 

2c-5 
c(2c  +  5) 

23. 

a(aj+2) 
a:(a:-7) 

29. 

a  — 6  -j-c  —  c? 
a  -|-6  — c  — d 

ALGEBRA. 

Art.  150 

;  page  85. 

2.     ""-^ ' 
3x  +  7 

^      m-1 

6m  — 5 

g      6  m  —  n      {.     3cc  —  2 

5m  — In             x-\-3 

3    5a  +  7 
a-2 

5.       ^  +  1    . 

7     a;  +  2           9     22/-3 
a;-3              '    2^-5 

10-    o"^. 

-a-1 

Art.  155  ;  pages  87  and  88. 

5.  s^-xy  +  y^  +  ^—'        10.    2m^-5mn-\-7n^-- — 

x-{-y  2m  — 3n 

6.  2x  +  Q-^^.  11.    a^      '^  3a-5 


a; -3  2^2 -a- 3 

7.  a-2+    ^^^-^  .  12.    ^+1+    .^  +  ^     » 

8.  30^-5-^^.  14.    2a;-3 ^^±-5 

4a;-l  3a^-2a;4-l 


Art.  156  ;  page  89. 

3     (^+^^  g      2ab  ^^    a^-Sx" 

x  a-\-b 

M     gy^  -\-4x  —  l  g  6xP  —  x  ^c 

x-i-S      '  '  2aj+l* 

^     5m^— 2m7i— 4n^     -^  a^4-6^  -^ 

3m+n  a—b 

6.  §^zif.  11.  _ly_,        17. 

8  a;-)/ 

7.   -^^.  12.  ^2!^.  18. 

m  +  w  m-\-n  a^  +  3a;  — 2 

13.  ^^'. 
a  +  6 


x-2 

2n^ 

w?  +  mn  +  T? 

9a^ 

2a;-l 

6a;y 

2y  —  x 

x'^-h^-l 

ANSWERS.  9 


Art.  157  ;  page  91. 

(a  +  3)(a2-4)'    (a  +  3)(a2-4)* 
7  «^  +  a^  + 1  a;  4-1 


g     m?i  (m^  —  n^)     m?  (m  +  n)  mn^ 

n  (m^  —  n^)       n  {m^  —  n^) '    w  (m^  —  n 

g     2(a^+c^-&  +  ci6-+^')     3(a^-ft-^6+a6^-6-^)     4(a^- 

rt.4  _  7)4  »  „4  _  7.4  '  ^4  _  ^ 


11. 

12. 


2a^b m^  —  n^ 

2  a  (a  —  b)  (m -{- n)     2  a  {a  —  b)  {m -\- n) 

ar^-9 07^-1 

(a;  _  1)  (a;  _  2)  (a;  -  3) '    (a;  -  1 )  (»  -  2)  (a;  -  3) ' 

a^-4 

(a;-l)(»-2)(a;-3)* 

Art.  158 ;  pages  92  to  97. 

g     12a;+7  g     3mV  — 4  q     4a6  — 6  — 4a^ 

36      '  *       6m^n^    *  •  *  12a^b 

M     6a  — 5b  -     5 6-  -f  4 g^  ^q     J_       ^^^      m^ 

lOa'ft^  *  '       120a6    *  *    15*  *    42 

24    *  *        24     '  *      18a;2  • 

l« !_  ^-     4  bed  +  6  acd  —  3  abd  —  2  abc 

60  '  4:8  abed 

17.   -?—  20.   ^^±A'.  23.    ^^. 

18  5  2^        4a6  g^  3m^-{-n^ 

'    e  +  x-a^'  '    a^-b""'  '    (771+71)  (m-w)' 

19.   !^ 22.      ^  25.    5 

a^  +  15a;  +  56  x  +  y  (ci  +  3)(a-2)2 


26.     ^^ 


10 

27.  ^^+i. 
a  —  b 

28.  0. 

4a^ 


x^-y^ 


30 

2 

34. 

Ax 

x{4:a^- 

1) 

a^  +  3x-10 

31. 

iab' 
c^-b' 

35. 

2. 

32. 

{\-xy 

36. 

0^-2 
a.(a;2-l) 

33. 

0. 

37. 

0. 

38.  (^^  +  ^)^ 40         ^(^-^)      . 

39.   1^-1^^ 43.     ^^-^^ 


(x  +  1)  (x  +  2)  (ic  -  3)  ab{a  -  b) 

44.     ^9a-l  ^g        2m +n  ^g       2  a 


6(a— 1)                     m{m^  —  n^)  a-\-b 

45.   -A_.  47.    L_ 49.       ^ 


9a;-a^  (aj+2)(a;+a)  a;2_i 

50.    —-^ 51.   0.         52.    -  ^ 


a^  -  5a;  +  6  (a;-2)  (aj-3)  (07-4) 

Art.  159  ;  pages  98  and  99. 

4.   1.  8.-^.  12.    ^^-^'.      16.   i±^. 

a  10  ax  +  &a;  a; 

5     1.  9   3^-1  iQ        ^y  17.    —^ — 

aa;— 2  a       18. 


6.   -^.  10.  ^-^-^^.    14. 


»• 

-2 

a^- 

-aj-20 

a^ 

a  - 

-26 

4a;2/  a^  a  +  1 


x—y+z 
19.    a^  +  a;?/. 


7.   a«.  11.  ^nl-^         15.   ^^±1^ 

a^  a^+2a;        20.    1. 

Art.  160;    pages  100  and  101. 

3.  —^—.  4.   ^V,  5.   «  +  6, 

106%V  '    oma;  '   a4-4 


ANSWERS.  11 

»     X'  -i^x  —  20  Q     m?  —  mn  +  n^         ^-     Sx  — 2y 

a;  —  2  m^  —  7^71  *      x  +  y 

12     ^^  '^^.  13     2a;  —  3y 

*     aH-6  *    2x-\-^y 

Art.  161 ;  pages  102  and  103. 

4.  x-1.  10.  ^!i±i?.  16.  °-^-''- 

6.  -i^.  11.  ^ii5.  17       1 


a  +  6  a;  +  l  l+ic^ 

6.  o?-x-\-l.  12.   ^.^Ili?.  18.   ^2jzM, 

7.  a-l.  13.   ^^^.  19.        ^ 


a  3a;  +  3 

8.  a?-2xy,  14.   ^~^,  20.    1. 

xy 

9.  ^.  15.  X.  21.    ^-« 


a;-f-6  x-\-2a 

22        ^^  23     2  (^  —  ^) 

•   a2  +  62*  (m  +  w)^' 

Art.  162 ;  pages  104  and  105. 
I     hx  —  a  o       m  —  1  g  "^ 


a;2  3m -15                        (1+a;)* 

4      ah-\-h\  5     __E!zi1__. 

am  +  an  *    2  (1  +  a;)(l +a^) 

g     2(a4-6)  9     a_^.                    -lo        5a;2 

7.    ar^  + 1+1  10.        ^ 


1  +  a^  ax  +  6 


8.   _!-,  11.        * 


A<V. 

3-3ar' 

13. 

2w-4 

3/1 

14. 

a;2  +  a;3/. 

12 

ALGEBRA. 

»^^- 

19. 

x'  +  y' 

23.   3. 

-  --? 

20. 

x-2 
x-6 

24.     ^^  +  ^  . 

{x-2y 

17.   ^^-l 
4a;* +  1 

21. 

a  — 36 

25.   ^'^''- 
a 

i«       2aa;2 

22. 

x^-f 

26.   30^  +  32/ 
x-7y 

oy             a6  +  6c  +  ca                     ^g 
(a4-6)(6  +  c)(cH-a) 

2x^-2 

28.        ^     . 

30. 

a^  +  a^^^  +  d* 

l  +  9aj  a6(a-6)^ 

Art.  174 ;  pages  109  and  110. 


3. 

14. 

8. 

2. 

13. 

2. 

19.   0. 

24. 

2. 

4. 

-5. 

9. 

3 

8* 

-If 

14. 

2 

3* 

-2. 

20.    1. 

25. 

1. 

5. 

3. 

10. 

15. 

21.    2. 

26. 

2. 

6. 

-3. 

11. 

1. 

16. 

16. 

22.    2. 

27. 

-8. 

7. 

-8. 

12. 

4 
3* 

18. 

3 

2* 

23.    -4. 

28. 

4 
3' 

29.   4. 

30.    -3. 

Art.  175  ;  pages  111  to  115. 

2. 

-6. 

7. 

-If- 

13. 

-2. 

18.    7. 

23. 

-2. 

3. 

2 
3* 

8. 

2 

7 

14. 

1 
3* 

19.    -lA 

.  24. 

2 

3* 

4. 

2 
3' 

9. 

-2f 

15. 

5. 

20.    -5. 

25. 

__1^ 
2* 

6. 

3. 

10. 

56. 

16. 

-3. 

21.   4. 

26. 

2 

3' 

6. 

5, 

11. 

2. 

17. 

1 
2* 

22.    -5. 

27. 

1. 

ANSWERS. 

13 

30. 

7. 

34.  - 

7.       38. 

5. 

42.  -4. 

«.  |. 

31. 

-ItV 

35.  - 

2.       39. 

3. 

43.  -  7. 

«■!■ 

32. 

2. 

36.  - 

1-  « 

0. 

44.  -2|. 

48.  - 

1. 

33. 

1 
2* 

37.  - 

1-  "■ 

1. 

45.  -  If, 

«.|. 

Art.  176  ;  pages  116  and  117. 

3. 

Sc—d 
2a-{-b 

8. 

2  m' 

13. 

3a-3. 

18. 

mn. 

4. 

5a 
2  6* 

9. 

a-b. 

14. 

n 
2 

19. 

7  a. 

5. 

a  +  1 
a-1 

10. 

a-^b. 

15. 

a 
b 

20. 

a 
3* 

6. 

Sb  +  4a.     11. 

a  —  b 
2 

16. 

I2a\ 

21. 

a 
3  6* 

7. 

5 

12. 
23. 

b-2c. 
1 

17. 

1 
a-\-2 

24.   n. 

22. 

Sb^ 

a  ' 

2{a-j-b) 

Art.  177  ;  page  118. 

4.  2.      6.  .7.    8.  8.      10.  0. 

5.  50.     7.  5.     9.  -.04. 

Art.  179 ;  pages  120  to  131. 

7.  35,  24.       10.  A,$30;B,$60. 

8.  A,  60;  B,  15.  11.  116,  91. 

9.  23.                        12.  7  and  10. 
13.    12  oxen,  24  cows.                        14.  47,33. 

15.  Wife,  $864  ;  a  daughter,  $288  ;  a  son,  $144. 

16.  A,  $18;  B,  $48;  C,  $4. 


2. 

.8. 

3. 

-8. 

4. 

30. 

5. 

20,  14. 

6. 

120. 

14  ALGEBRA. 

17.  Infantry,  2450  ;  cavalry,  196  ;  artillery,  98. 

18.  A,  62  ;  B,  28. 

19.  On  foot,  880  ;  by  water,  1540  ;  on  horseback,  616. 

21.   - — ,      ^^^  •       22.    22,  23,  24,  25.       23.    29,  14 

1  4-  mn     1  4-  ^^^^ 

24.   A,  $14;  B,  $13;  C,  $11;  D,  $9.         25.    115. 
26.    120,  60,  20,  5.    27.  A,  59  ;  B,  23.    28.  22,  23.    29.  36. 

30.   A,^^fr"^^  B,^(^-^).  33.    8tY. 

m  —  n  m  —  n 

qi  gn^  an  a  oa     11. 

•     n^j^n  +  l'    iv'  +  n  +  l'    T^  +  n  +  l  '    12* 

35.    144  sq.  yds.         36.   A,  $35  ;  B,  $38.         37.  -^. 

38.    ^ 40.    2  dollars,  20  dimes,  4  cents. 

ah  +  hc-\-  ca 

41.    17  two-penny  pieces,  36  farthings.  42.    $2.75. 

43.   Worked,  20;  absent,  16.       44.    $58.  45.   "IZL^. 

c  -f"  1 
46.   48  minutes.  48.    84.  49.    93. 

50.    30  bushels  at  9  shillings ;  10  at  13  shillings. 

51.   Gold,  3377  oz.;  silver,  783  oz.     52.  36.     54.  8f  miles. 

55.      ^^  .  56.    A,  12  miles;  B,  14  miles. 

h  —  a 

67.  $1200  in  5  per  cents  ;  $2000  in  6  per  cents. 

gg^      100a   ,  59^    \00{a-p)^  ^^     28,  9. 

rt  + 100  pr 

62.    82,  31.  63.    12,121  men  ;  110  on  a  side  at  first. 

64.   ^^-=^^   ^-^-±^-     65.   -1.      66.  51.      67.  MK^. 
&  +  1      6  +  1  13  pt 

68.  Picture,  $5.28;  frame,  $3.96.  69.   2. 

4 

71.  5^^  minutes  after  7.  73.    27^^  minutes  after  5. 

72.  43i2j-  minutes  after  2.  74.    5j^  minutes  after  1. 


ANSWERS.  15 

75.  7.  77.   -Smiles. 

64-c 

76.  7  dimes,  13  half-dimes.     78.    16^  minutes  after  6. 

79.  5-j^  minutes,  or  38^^  minutes  after  10. 

80.  First  kind,  ^^iiZl*)  ;  second,  M^^nil. 

a—b  a—b 

81.  10  A.M.  82.  27^^  minutes  after  4. 

83.  48.  85.  Horse,  $180  ;  carriage,  $95. 

84.  $6480.  86.   A,  52  miles  ;  B,  55  miles. 

87.  ^^^  +  ^^  +  ^  cents. 

88.  A,  11  days  ;  B,  22  days  ;  C,  33  days. 

89.  A,  $750;  B,  $500.         90.    48  minutes  after  9. 

91.  A,  3  days  ;  B,  4  days  ;  C,  5  days. 

92.  18  yards,  15  yards.         93.   5^. 

a  —  G 

94.    $2000.  95.    $1400.  96.    $15,000. 

97.    16/y  minutes  after  8.        98.    Greyhound,  72  ;  fox,  108. 

Art.  184 ;   pages  134  and  135. 

3.    x=5,  7.    x  =  -3,     11.    x  =  -4.,      15.    x=    8, 

2/  =  -2.  y  =  6.  2/  =  -5.  2/  =  10. 


8.    x=7,         12.    x  =  -2, 
4.    .T  =  4,  2/  =  -l.  y  =  ^' 


2/  =  3. 


9.   x  =  K         13.    x  =  -t 


16.    x  =  -l, 

y  =  -3. 


2  3  ^_2 

5.  x  =  o,                 2/  =  -3.  y  =  -2.  ^^'    ""-      3' 

?/  =  2.                              o  9  5 

10.   x  =  -|,  14.   a;  =  -|,  2/  =  -^ 

6.  X-.V2,                     2.  3. 
y=7.                  ^5  •'2 


.6 

ALGEBRA. 

Art.  185; 

page 

^  136. 

• 

2. 

a;  =  5, 

5. 

x=l. 

8. 

05=-l, 

11. 

9 

05  =  -, 

2/  =  2. 

1 

2/  =  -2. 

5 
3 

3. 

0^=3, 

6. 

1 

9. 

1 

1 

2/  =  -l. 

2/  =  -l- 

12. 

05=--, 

y=x 

.=. 

2/  =  -3. 

4. 

aj  =  -2, 

7. 

.T  =  -3, 

10. 

13. 

05  =  -2, 

2/  =  -2. 

2/  =  -2. 

3,  =  2. 

2/  =  19. 

Art.  186; 

page  137. 

2. 

05=2, 

6. 

a;  =12, 

9. 

a;  =  —  3, 

12. 

05=1, 

3. 

2/  =  3. 

1 

.  =  --, 

7. 

2/  =  -3. 

a5  =  -l, 
2/  =  4. 

10. 

1 

.  =  _-, 

2 

13. 

1 

2 

"'  =  -3^ 

4. 

£C=  5, 

2/  =  -3- 

2/  =  l. 
^      2 

2/  =  -2. 

6. 

«7 

a:  =  -2, 
2/  =  3. 

8. 

05  =  4, 
2 

11. 

1 

2/  =  2. 

Art.  187;   pages  138  to  143. 

2. 

ic=10, 

6. 

05=2, 

11. 

x  =  -2, 

15. 

05=3, 

2/-    5. 

2/  =  3. 

2/  =  -4. 

2/  =  -4. 

3. 

a;  =  24, 

7. 

05  =  -. 3, 

12. 

x=2, 

16. 

1 

05  =  -, 

2/  =  -18. 

2/ =.08. 

1 

3' 

4. 

a;=-16, 

8. 

05=18, 

^  =  i- 

3 

6. 

2/ =  -12. 

3 
^=2' 

9. 

y=  6. 

05  =  4, 

2/  =  9. 

13. 

05=60, 
2/ =  40. 

17. 

05  =  3, 

2/  =  -2. 

4 

10. 

05  =  -2, 

14. 

05=1, 

18. 

05  =  -3, 

^  =  -3- 

2/  =  3. 

2/  =  -2. 

2/=ll. 

ANSWERS.  17 

19.  0^  =  ^,         21.    0^=1,         23.   a;  =  2,         25.   aj  =  -— , 

2^  =  T'  2/  =  -— • 

22.   a;  =  -6,     24.   «  =  13,  ^ 

20.  a;  =12,  ^  =  -5.  2/=    3. 

2/=   6. 

27.  a.  =  i^±i^,      29.   x  =  '-^^,         31.   a,^^^  +  ^^, 

17  a-h  ad-{-hc 

2  6  —  3  a  c  —  ad  cm  — an 

^  17  ^       a-h  ^      ad  +  hc 

28.  .T  =  ^^Lzi^,      30.   0.'=— 5^-,      32.   a^  =  i^^£il^, 

ad  — he  an-{-hm  mn'—m'n 

an  —  cm  ap  m'p  —  mp ' 

w  = •  y  =" •  2/~ —  — 

ad  —  he  an  -|-  6m 

rtQ  ac(bm  +  (^^0  6d(cw  —  am) 

jOo.   ic  = )  y  —  — ^ ' 

ad-\-hc  ad-^hc 

34.   x  =  ah,  ^    x  =  -^,  47.   x  =  K 

y  =  a-\-h.  m-\-n  2 

m^  —  n^  1 

36.   0;  =  -,  a  4 

a 

J  Ai        _  1  x  =  m  +  n, 

2/  =  _.  41.    a:-—,  2/  =  mH-n. 

36.    x  =  a%  y  =  ^'  49.   a.  =  |^:z^, 

^  <^  bn  —  dm 


y  =  aZ)^. 


42.   x=^{a  +  h)\  y^bo-(^, 

cm  —  a/i 


n 


37.  a^  =  ^,  y  =  {a-hy. 

44.   .  =  4,  50.   .  =  1 

2/  =  2.  3 

♦  w  =  -. 

4 

38.  a;  =  a  +  6,  45.   a;  =  -5, 

y  =  a-h.                    2/=3.  51,    a;=i, 

39.  x  =  a,                 46.   a;  =  -2,  2/  =  — 
y  =  b.                         y  =  —l,  m 


.8 

ALGEBRA. 

Art.  189 ;  pages  145  to  147. 

3. 

Xz=S, 

10. 

a;  =  -7, 

16. 

x=lO, 

21. 

x=-i, 

y  =  -h 

2/  =  -2, 

2/=    2, 

3' 

z=0. 

;2  =  1. 

2^=    3. 

'=!■ 

4. 

x=-2, 

11. 

a?  =  8, 

17. 

aj  =  -24, 

1 

2/  =  3, 

2/  =  -3, 

2/  =  -48, 

2;  =  -• 
4 

z  =  l. 

2  =-4. 

2  =  60. 

22. 

1 

aj  =  -, 

5. 

2/  =  2, 

12. 

— 1' 

18. 

"4. 

0  =  -4. 

3 

a.  =  -i, 

6. 

x  =  2, 

'•i- 

6' 
1 

1 

2/  =  -l, 

2=-2. 

'=-!• 

2/  =  ^, 
z=-^. 

23. 

w=4, 
a;  =  5, 

7. 

a;  =  23, 

13. 

aj  =  — 5, 

10 

2/  =  6, 

2/=    6, 

2/  =  — 5, 

19. 

w  =  —  7, 

2=7. 

^  =  24. 

^=-5. 

a;  =  3, 

24. 

x=K 

8. 

x  =  -2, 

14. 

a;  =3, 

2/  =  -5, 
;?  =  1. 

2' 
1 

^  =  3, 

2/  =  4, 

4 

2/  =  -3, 

z  =  1. 

2  =6. 

20. 

2=-l. 

9. 

aj  =  -4, 

15. 

a^  =  4, 

2/  =  4, 

25. 

a^  =  7, 

2/  =  2, 

2/  =  6, 

_4 

2/  =  -3, 

^  =  -5. 

;3=2. 

^  ~5' 

2  =-5. 

26.    a^: 

h''-\-c--  a' 

28.    a^  = 

=  3, 

26c 

y  = 

=  -S 

I 

c'^j^a'-V' 

2  = 

=  -1 

.. 

2/  = 

-^ 

2ca 

a^- 

^h'^-c^ 

29.   a!  = 

=  a, 

2;  : 

2  ah 

2/^ 

=  1. 

27.   aj: 

=  li 

y- 

=  1, 

30.    a.= 

=  abc 

? 

z  = 

1 

y  = 

=  ah  -f-  he  4-  ca, 

2* 

z  = 

=  a  +  6  +  c. 

ANSWERS.  X9 

Art.  190 ;  pages  149  to  157. 

3.  32,18.         5.    A,  30;  B,  20.         7.    24  and  18. 

4.  14,  7.  6.    A.  8.   A,  $96;  B,  $48. 

15 

9.    A,  48  ;  B,  18.      10.  ~     11.  A,  16  days  ;  B,  26f  days. 
1  y 

12.    38,  13.  13.  -•  14. 


5  a^  -  a  +  1 

15.  Better  horse,  $160  ;  poorer  horse,  $100  ;  harness,  $40. 

16.  First,  8  cents  ;  second,  7  cents  ;  thh'd,  4  cents. 

18.  30  cents  ;  15  oranges. 

19.  l^  bushels  at  60  cents  ;  26|  at  90  cents. 

20.  Income  tax,  $20  ;  assessed  tax,  $30. 

21.  120,  at  7  cents  each.  24.    10,  22,  26. 

22.  Length,  80  ft.  ;  width,  60  ft.       25.   42,  38,  32,  24. 

23.  A,  $45;  B,  $55.  27.    74.         28.   326. 

29.  A,  9f  days  ;  B,  16  days  ;  C,  48  days. 

30.  Length,  30  rods ;  width,  20  rods ;  area,  600  sq.  rods. 

31     h-\-  c  —  a     c-j-a  —  b     a-^h  —  c  «o     246 

2        '  2        '  2        * 

33.  Number  of  persons,  {''  +  ^)'^'' ; 

bm  —  an 

each  received  — ^ — — — ^  dollars. 

bm  —  ail 

34.  Whole  sum,  $1200;    eldest,   $400;    second,  $300; 

third,  $240;  fourth,  $260. 

35.  30  at  2  for  5  cents  ;  36  at  3  for  8  cents. 

36.    42.  37.    38. 

38.   A, ^JH^ days;   B,  ^^^^ ; 

mn  -{-np  —  mp  mp  -^  np  —  mn 


C. 


2mnp 


mp  -+-  tnn  —  7ip 


20  ALGEBRA. 

.f.     r\  .ad  —  hc     -1  ,  ad -{-he 

40     Current,  miles  an  hour :  crew,  ' 

'      ^bd  2bd 

41.  Going,  4  hours;  returning,  6  hours. 

42  759.  43.    First,  22  ;  second,  10. 

44.  65.  45.    First  rate,  6  p.  c.  ;  second,  5  p.  c. 

46.  $120  at  5  per  cent. 

47.  t^SZ^  dollars  at  ^^^  ^^  "  ^^  per  cent. 

m-—n  hm  —  an 

49.    15  miles  ;  b\  miles  an  hour.  50.   43. 

51.  40  miles  an  hour.  53.    A,  8  ;  B,  6. 

52.  A,  $13;  B,  $7;  C,  $4.  54.    58,43,14. 

55.  A,  $7;  B,  $22;  C,  $21;  D,  $16. 

56.  A,  $78;  B,  $42;  C,  $24. 

57.  A,  8  hrs.  ;  B,  9  hrs.  ;  C,  12  hrs. 

58.  $2000  in  ^  per  cents  ;  $1600  in  four  per  cents. 

59.  A,  16  ;  B,  12.       60.  Fore-wheel,  8  ft.  ;  hind-wheel,  12  ft. 
61.    A,  8  days  :  B,  12  days.  62.   A,  6  ;  B,  5. 

Art.  194;    page  160. 

4.    4a;^+4ar+5a^-f2ic+l.         6.    .T'^+8^+12a.'2-16a:+4. 
6.    ic*-6a^+ll^'^-6a?+l.  7.    4a^^-4a^-lliK2+6a;+9. 

8.  9a^-30a^  +  49a2-40a+16. 

9.  4a^4_^20a^-3a52_70aj4-49. 

11.  aj«-4a;^  +  10a;^-f  4.T2-20a;  +  25. 

12.  4a;«  +  12a^  +  9a^^+.4ic3  +  6a.'2  +  l. 

13.  9a^-12a36-26a262  +  20a63  4-25&^ 

14.  1 6  m*  +  8  mH^  -  23  mhi^  -  6  mn^  +  9  n^. 

17.  l  +  2a;-f  3a;2  +  4a^  +  3a;4  +  2i^  +  a^^ 

18.  9a^-12a;^-2aj*  +  28a^-15a;2_8.'»+16. 

19.  a;6-8a^+12aj^-t-10a^+28a^  +  12a;  +  9. 


ANSWERS.  21 

Art.  195;   page  162 

3.  a^-{-da^-h27x-\-27.          6.    a' -i- 12  a^b  + 48  ab^ -{-64  b\ 

4.  8ar^-12a^  +  6a;-l.         7.    27m«- 27m*+ Qm^- 1. 

9.  d'  +  Ua^b-{-15ab^-^125b\ 

10.  8a^-60a^2/+ 150a;/ -125/. 

11.  8a;»-36a;^  +  54ar^-27a;'. 

12.  216x^-\-10Sx'y-\-18xy  +  a^f. 

13.  27 m-^  +  135  mrn  +  225  w,?*^  +  125 n\ 

14.  27 ar^/  -  1 08  aV/  +  144 a^xy  -  64  a«. 
16.  if^-Sar^  +  Sx-^-Sa^-l. 

17.  a»-3a26  +  3a2  +  3«V_ea6  +  3a-6'  +  3Z>2_36  +  l. 

18.  a3H-3a-^-3a2c+3«62-6a6c+3ac2+63-362c+36c2-c3. 

19.  x«-6ar^+18a;^-32a^  +  36a^-24a;  +  8. 

20.  ic^  +  9a^  +  30a;^  +  45.'c^H-30a^+9x+l. 

21.  8x^-36x'-\-i2x*-\-9a^^21x'-9x-l, 

Art.  196 ;   page  164. 

10.  16-^S2x-\-2ia^-\-8x'-j-x\ 

11.  x'-lQx^-{-d6x^-2o(jx-\-2D6. 

12.  a^-15a^  +  90a3-270a2  +  405a-243. 

13.  a^4-10a^  +  40a3+80a-  +  80a  +  32. 

14.  x"^  -  12x^  -\-  60x*  -  IGOaf  -\-  240x^ -  ld2x  -\-  64:. 

16.  a^  -  Ua'x  +  QOa^a^  -  270aV  +  405a«^  -  243ar*. 

17.  81  +  216&  +  21662  4-96&«+166^ 

19.  a^-16i»^+9Ga;«-256a:^  +  256. 

20.  64a^-  +  192a^«6  +  240a852+160a^6H60a*6^4-12a^5'+6*. 

21.  16m''  -  96mV  +  216mV  -  216m%«  +  8171^. 

Art.  204 ;  page  169. 


3. 

a2-2a  +  l. 

6.    5-^3ff  +  xK 

9. 

2a'-5ab-\-8b'. 

4. 

2ar^-a;-l. 

7.    3a^-4a;-5. 

10. 

l-7«-2a^. 

6. 

S-2x  +  x^. 

8.    7^  +  1--- 
m 

11. 

a  —  b  —  c. 

22 


ALGEBRA. 


12.  x-2y  +  Sz.         13.  3ar^+5ar^-7.         14.  4c3-5c-3. 

15.  a3_2a2  +  5a  +  3. 

16.  2oif —  x^y —  xy^ —  2y^. 

17.  2a^-3a^+4a;-5. 

18. a6  H 

3  ^2 

19.  3x»-2a^2/  +  a;i/2-22/^ 


20.  1+^-^4-^. 

2       8       16 

21.  i-a-^-^ 

2        2 


22.    a_26-^^-^^ 


36^     36' 
2a        a^ 


23.    2a.  +  X_^  +  ^L 
2a;      IGa;'^      640.-^ 


Art.  207 ;  pages  172  aftid  173. 


2. 

214. 

10. 

21.12. 

19. 

5.5678. 

27. 

1.3229, 

3, 

523. 

11. 

.04738. 

20. 

4.1593. 

28. 

.43301, 

4. 

809. 

12. 

900.8. 

21. 

.83666. 

29. 

1.0541, 

5. 

5.76. 

13. 

.8253. 

22. 

.28284. 

30. 

.44721 

6. 

.497. 

15. 

1.4142. 

23. 

.37947. 

31. 

1.1547. 

7. 

.286. 

16. 

1.7321. 

24. 

.031623. 

32. 

.64550, 

8. 

.722. 

17. 

2.2361. 

25. 

.079057. 

33. 

1.1726 

9. 

1.082. 

18. 

3.3166. 

26. 

1.4444. 

34. 

.94868, 

35. 

.62361. 

36.    .42492. 

Art.  209 

;  page  175. 

7. 

a^  +  2x 

-4. 

10.    2x^- 

-3a;- 

-1.     13. 

x^-i-xy 

-2y\ 

8. 

y'-y- 

-1. 

11.    2a'- 

-  a  — 

5.       14. 

3a'-2ab-b\ 

9. 

x^-2x 

+  1. 

12.    2 -a; +  2 

ar^ 

Art.  213  ;  pages  178  and  179, 

2. 

31. 

8. 

2.02. 

14. 

.898. 

20. 

.3107. 

3. 

4.6. 

9. 

372. 

15. 

101.3. 

21. 

.7211. 

4. 

.88. 

10. 

21.6. 

16. 

.0534. 

22. 

1.077. 

5. 

123. 

11. 

.803. 

17. 

1.260. 

23. 

.6376. 

6. 

1.14. 

12. 

4.89. 

18. 

1.817. 

24. 

.8736. 

7. 

.098. 

13. 

.317. 

19. 

1.931. 

ANSWERS.  23 

Art.  214 ;  page  179. 
1.   2a;-32/.  2.    a^-x-{-l.  3.   a^-2a;-2. 

Art.  219 ;  page  182. 

21.  243.  23.    216.  25.    -243.  27.    128. 

22.  81.  24.    32.  26.    16.  28.    1296. 

Art.  224  ;  pages  184  and  185. 

10.    4mi  11.    QaS.  14.    5aj~3.         15.    Sahj. 

17.    a4_2_|.a-*.  18.    a'-a^.  19.   oj-^-l. 

20.   X  ''^Sx-^-4:X  \  21.    18a26-  +  10  +  2a-26-2. 

22.    6a.'2-7a;3-19a;^  +  5a;H-9a;^-2a;i 

23.  2x-^y-10xy~^-\-8x^y-\        24.   2 -4a~^a;^ +  2a"^a!3. 

25.    18ab~^-23+a''h-\-6a-^b\ 

Art.  225 ;  pages  185  and  186. 

5.    3c-l         6.   m'A         7.   a;ii         11.   J-{.ah^  +  b^. 

12.    a"^-a~^  +  l.         13.   ic-^-9-lOa;.         14.   a;"^  +  l. 

15.  m^-2m^n*+rii  17.   a^  -\-ah^-^bK 

16.  x~^y'^  —  x-^y-^  —  x-^y-\        18.   m~^  — ri-^  +  m^w-^ 

Art.  227;   page  187. 
10.  f.         13.   c"^.         14.   w  1.         15.   x~K         16.   a-. 

Art.  229;  pages  188  and  189. 

8.  Sx'^-2x  ^-1.  16.   x^-Sx^-\-2xK  21.   a'^. 

9.  2x^-{-x-4:XK  18.    a^*-"*.  22.   a;. 
10.    a^6~^-2  +  a-*6i          19.    x'^^'^K  ^, 
15.    2y-^^-y-K                      20.    a:«   ^  2a^ 


24  ALGEBRA. 

24.  cfil^-c^l^,      26.    -^.  28.   ^^^. 

25.  1+a;.  27.    (1  -  Sa^  +  a.-^)"'.     29.    (l+ar')i 

Art.  235;  page  191. 

9.   "V^.  10.  V2a.  11.   Vlwh^. 

12.  V5^.  .  13.    V^. 

Art.  236;   page  192. 

12.  bxy^xy''-2x'y.  15.    (.t-3)V«. 

13.  ^ah^'2aJ)'  +  bh.  16-    (2.^•^-3)V5. 


14.    (a?  +  2/)  Va;  —  2/.  17.    (m  — 97i)V3m. 

19.   iv«-     21-   ^V21-     23.   |-?/6i.      25.    1V7. 


77 


20.   Iv30.   22.   ^V3-    24.   i^l.5.      26.   :^,Vl06cci. 


27.   ^Vl4^.  28.     2(^  +  ^)  •         29.   ^j^^^- 

Art.  237;  page  193. 
10.  V^f^l.     11.  Vl^^.     12.  \^r^-      13.  V4-4a^. 

Art.  238;   pages  194  and  195. 

3.  5V3.         9.    (2a+3&)V^.     14.   5^2  +  ^-^18. 

4.  7V6.       10.    9V3-7V5.      15.    9^3.    ^ 

5.  3V5.       11.    9^2.  16.    6aV3^. 

^'    f^'         12.    ^V^-  ^'^-    (3ci2-263)V^rr26. 

7.    20  V2.  5  jg^    10^2-2-^3. 

8-   ^-       ^^-   ^V6.  19.    (a-4)V7^. 


15  36 


20.    2Va^-/, 


ANSWERS.  25 

Art.  239 ;  pages  195  and  196. 

2.  ^^4,   ^27.  4.    ^625,    ^216,  ^49. 

3.  ^"125,   ^16,  </9.  5.    ^32^,  VWb^  a/64?. 

6.    a/^S  ^^,  a/^. 

8.    i/2,  9.    </5.  10.    V3,   </7,  ^4. 

Art.  240 ;  pages  196  to  198. 

3.  6V7.  7.  -^3.  11.  12^/288. 

4.  75V6.  8.  ^432.  12.  ^xyz. 

5.  30W5.  9.  a/^¥^.  13.  ^3. 

6.  a\/P?.  10.  2aW^.  14.  ^72^. 

17.  x  +  -^x-6.  23.    4  +  2V10. 

18.  4-5V10.  24.    28  +  8V42. 

19.  2a;-7V3^-12.  25.    11  -  20  Vl5  4- 5  VlO. 

20.  Sa-lO^ab-Sb.  26.    21-12V3. 

21.  x-y-\-2^-z.  27.    147  +  60V6. 

22.  2-3V^T^.  28.    l  +  2aVl-a^ 

29.    2a-2V^^=T^        30.    1.        31.   2.        32.   6x  +  i9. 

Art.  241 ;  page  199. 


2.   3V2.  6.    41  8.    ^54.  10.   f/^- 

6.    ^24.  7.    ^g.  9.    ^32^.        11.    ^J/18^. 

Art.  242  ;  pages  199  and  200. 
6.    a^xV^.  8.    192ajV3^.  10.    S\a'bx</b^. 

6.    3V2.  9.    2^2.  11.    18ab^/S^'. 

17.    </3^.     19.  ^^/'^^n.      20.  V3.      21.  ^CW.     22.  V2. 


26 

ALGEBRA. 
Art.  243 ;  page  201. 

4. 

^V5:r2/.     5. 

3^^3a.     6.    j-^V2af, 

7.   0^^3«^ 

3. 

12-4V2 

7 

5  +  2V3. 
2V6-5. 
a4-2V'a6  +  & 

16  +  7V10 
13 

a^  —  x 

Art.  244;    page  202. 

9.   -^-^- 

10.  a;+l- 

11.  2a'-: 

12.  -^-^ 

13.  ^  +  "^^ 

a 

-Va+1 

4. 

-Vo^  +  2x. 

5. 

L-2aVa2-l. 

6. 

-t-a;Va;^-4 
2 

7. 

x^-or^ 

8. 

:.                    14.   Va*- 

l-a\ 

Art.  245;   page  203. 

2. 

7.243.              a 

1.    3.365.              4.    .101. 
Art.  249;   page  205. 

6.    .269. 

3. 
4. 

-  8  V6.       6. 

—  ax.          6. 

12Va&.             7.    5-5^ 
-2V-15.      8.    46. 

/-I.     9.    12. 
10.    60. 

11. 
12. 
13. 

l_4V-3. 

-11-4V6. 

2. 

14.  a'  +  b. 

15.  y'-a^. 

16.  0. 

Art.  256;   page  208. 

19.  2V2. 

20.  V2. 

21.  V3. 

4. 
5. 
6. 

V5-2.   ^ 
3  4-V7.     ^ 

V5  -  V3. 

14.    3V5-V5 

7.  2V2  4-V7.        10. 

8.  3-V3.               11. 

9.  V15-V5-         13. 
J.                   16.    Vm+n- 

3+V5- 

3V2  +  V5. 
5  +  2V2. 
-  Vm  —  n. 

15.    7-3V2. 

17.    ^a  +  x-\ 

-Va. 

ANSWERS.  27 

Art.  257 ;    pages  209  and  210. 


3. 

2. 

8. 

2 
3* 

13. 

-3. 

18. 

-3. 

23. 

13 

4* 

4. 
5. 

4. 

6. 

9. 
10. 

4. 

81. 

14. 
15. 

2a. 
-1. 

19. 
20. 

25. 
12. 

24. 
25. 

a6 
a  +  6 
-6. 

6. 

5. 

11. 

4. 

16. 

4. 

21. 

3. 

26. 

3. 

7. 

-2. 

12. 

8. 

17. 

5. 

22. 

a 

27. 

3a-l. 

Art.  259;  page  212. 

3.  ±3.  7.  ±J^.  11-   ±3.  15.  ±^ 

4.  ±5.  8.  ±1.  12.    ±2.  16.  ±2. 

5.  ±|.  9.  ±V19-  13.   ±Vm^.  17.  ±1. 

o 

6.  ±2V^^.   10.  ±2.  14.   ±y/2.  18.  ±i. 


Art.  262  ;  page  215. 

3.  1,-5.     5.  4,3.      7.  2,-|.        9.  |,1.  11.  1,-| 

4.  4,1.         6.  2,-3.8.   -2,-i.  10.  3,-i.  12.  |,-| 

2                    4  o       o 


Art.  263 ;  page  216. 

3.  ,,.U.     5.1.1         ,.  ..-H.     ..l.|. 


28 

ALGEBRA. 

Art  265 ;  page  218 

3. 

-,,i. 

6. 

-2, 

'I-  »•  '■ 

2_ 

7 

-i'-i 

4. 

■•  -!• 

7. 

1 

2' 

i  "-^ 

4, 

5 
3' 

-4-i 

5. 

6,  -3. 

8. 

5,  - 

-i-  "■ «. 

— 

1 
5* 

2'    3 

15. 

1    1 
3' 5* 

'»!■ 

2 
5* 

Art.  266  ;  pages  218  to  221. 

1. 

1         1 

2'       6* 

13. 

-10  ±^78. 

25. 

4,0. 

2. 

-1,  -4. 

14. 

'■1- 

26. 

-2,12. 

'65 

3. 

■■-!■ 

15. 

'■!■ 

27. 

'■^■ 

4. 

5    15 

2'   4* 

16. 

'•  -1- 

28. 

'■'f 

5. 

13,  -2. 

17. 

7       5 

2'      2 

29. 

3,-2. 

6. 

1     1 
2'T4* 

18. 

^■f 

30. 

».-!■ 

7. 

'•f 

19. 

'■i- 

33. 

a +  5,  a  —  b. 

8. 

3        1 
2'~2" 

20. 

■•1 

34. 

a,  —6. 

9. 

-•-f 

21. 

'23 

35. 

l,a. 

10. 

5,  -3. 

22. 

1,  -18. 

36. 

-16,  -2c. 

11. 

4±2V3. 

23. 

-3,51. 
'  22 

37. 

m^  —  m^ 

12. 

'■-I- 

24. 

2. 

38. 

5    d 
a  G 

ANSWERS.  29 

39.  2p,  -  bp.  41.    2^,  l-  43.   ^,  t 

4:     2  ha 

40.  -^,  — *•        42.   m, ^.  44.   o,  i. 

2         3  wi  + 1  a 

45.  a  +  6,   46.  50.   ^^±^, -^. 

c       «.  +  & 

46.  (a +  6)2,  -(a -6)2.  51.    -a,  -6. 

47.  9m,  —  m.  52.    a  — 6,  —  a  — c. 

AQ^        o«  K«2a  — 6        3a  +  26 

4o.    a,  —2a.  Do.    , ' • 

ac  he 

49.    -a, -2a.  54.   ?dc±,?L=^. 

a  —  b   a-\-b 

Art.  267  ;  page  222. 

3.  2,   --•  5.    5,  2.  7.    -1,   -i. 

2  6 

4.  5,  -1.  6.    -2,  5.'  8.    2,  i. 
3                                             5  5 

J,     5         1  11     «     2 

^-    2'   -2  ^^'    3'   3* 

10.    --,--•  12.   -,   -- 

Art.  268;  pages  224  to  227. 

3.  10  barrels,  at  $17.50.        5.    21,6. 

4.  9,6.  6.   Length,  125  ;  breadth,  50. 

7.  14,  5.  12.    16.  17.    9. 

8.  7,8.  13.    16  barrels,  at  $6.       18.    15,9. 

9.  16,  10.  14.    13,  6.  19.    3712. 

10.  5,3.  15.    3  inches.  20.    $80  or  $20. 

11.  7,  8,  9.  16.    $30.  21.    20. 

22.  Area  of  court,  529  sq.  yds.  ;  width  of  walk,  4  yds. 

23.  36  bushels,  at  $1.40.  24.  84. 


30  ALGEBRA. 

25.  Larger  pipe,  5  hours  ;  smaller,  7  hours. 

26.  $2000.         27.    5.         28.    6.         29.    343  cubic  inches. 

30.  First,  14,400  ;  second,  625  ;  or,  first,  8464  ;  second,  6561. 

31.  38  or  266  miles.  32.    70  miles. 
33.    100,  at  $15  each. 

Art.  270 ;  pages  229  and  230. 

4.  ±3,   ±4.  7.    ±i,  ±iv5.        10.    ±1,   ±2. 

2        o 

5.  -^3,   --^23.      8.    ±1,    ±^.  11.    2,   -3. 


6. 

1,   -2. 

9. 

^'-i- 

12.    4,  ^49 

13. 

243,   -28 

^784. 

14. 

(± 

„,  (,0.. 

15. 

"•  (f  )'■ 

17. 

1     /2V^ 
4'    W  * 

■»•  '■  ?• 

16. 

■••  (I)'- 

18. 
21. 

25,  A. 
'   16 

-32,  2-i 

20.    ±8,(^- 

Art.  271 ;  page  232. 

4.  2,   -2,  3,  7.  11.  8,   -2,  3±V110. 

5.  2,   -2,   -3,   -7.  12.  6,   -9. 

6.  1,  9,  5±2V2.  13.  I   -|,    ~^y^^> 

7.  ±2,   ±V11-  14.  -2,   -^15. 
b.  3,   --, 15.  0,  2,  -,    -. 

9.  2,   -3,  4,   -5.  16.  1,   -1,   -6,   -8. 

10.    1,  2,   -5,  8.  17.    0,   -5,  -,    -— . 

3  3 

18.    a  +  6^  a4-3-\/36"^ 


ANSWERS.  31 


Art.  274;   page  234. 


Note.   In   this,  and  the  three  following  articles,  the  answers  are 
arranged  in  the  order  in  which  they  are  to  be  taken ;  thus,  in  Ex.  2, 

the  value  a:  =  1  is  to  be  taken  with  y  =  —2,  and  x  — with  y  =  —. 

25  ^      25 


2. 

a^  =  i,  -^; 

6. 

.T  =  3,  -4; 

11. 

.  =  3,-1^; 

25 

.>    46 

2/  =  4,  -3. 

13' 

-,    62 

2/  =  ~^9  :;7* 

2/  =  2,  — 

25 

7. 

05=2,3; 

^           13 

3. 

x=l,  -8; 

y  =  -3,  -2. 

12. 

.  =  1,-1; 

2/=-8,  7. 

8. 

a;  =  5,  -2; 

2' 

7/  =  2,   -1. 

4. 

a;  =  9,  -6; 
2/  =  6,  -9. 

^  =  -2,5. 

13. 

-».f^ 

5. 

.=  2,-1; 

9. 

a;  =  4,  2/  =  6. 

'-.-f 

A      5 

10. 

a;=6,  -4; 

14. 

a;  =  4,  -2; 

2,  =  4,-. 

2/  =  -4,  6. 

2/  =  8,  -1. 

Art.  275 ;  page  237. 


4. 

a;  =  3,  -2; 

10. 

x  =  b^  — 3  ; 

15. 

05=7,  -12; 

2/  =  -2,3. 

2/=3,  -5. 

2/  =  -12,  7. 

5. 

ic=9,  —3; 

11. 

a;=2,  1; 

16. 

05=15,  -3; 

2/  =  3,  -9. 

2/=l,2. 

2/  =  -3,  15. 

6. 

a;  =  -3,  -7; 

.  =  8,  -y, 

15        c 

2/  =  V  ~^- 

2/  =  7,  3. 

12. 

17. 

05  =  -l,    -4 

7. 

a;=2,  -3; 

2/  =  4,  1. 

2/  =  -3,  2. 

2 

8. 

ic=  ±4,  ±3; 
y=  ±3,  ±4. 

13. 

05=  ±7,  ±6; 
2/=  ±6,  ±7. 

18. 

05=6,  -11; 
2/=ll,  -6. 

9. 

a;  =  3,  -7; 

14. 

05  =  3,  -7; 

19. 

05  =  7,-9; 

2/  =  -7,  3. 

2/=7,  -3. 

2/  =  -9,  7. 

32  ALGEBRA. 

Art.  276;  page  239. 
2.   x=±7,  ±1^2;  7.   x  =  ±3,  ±|; 

2,  =  ±2,  t|v2.  2/  =  ±5,  ±^- 

3^  ^  =  ±1,^19. 

3  9.    x=±2,  ±|fV31 

4.   x=±2,  ±|V5;  2,  =  ±3,  T^V31. 

i/=t3,  ±|V5- 


5.    a;=±2,  ±ivi^; 


10.   a;  =  ±2,  i^Vll 


2        5  11.    a5=±l,   ±2; 

6.    x=±3,  ±2V2;  y=±-,  ±-- 

2/  =  ±l,   ±V2.  ^'       ^ 
Art.  277;  pages  242  and  243. 

6.    ^  =  -5,  --;  10.    x=2m,  -m; 

^  y  —  m^  —2m. 

y=l,  --• 

3  11.    a;  =  3,  2,  -3±V3; 

6.  a;=l,  8;  2/  =  2,3,  -S^V^- 

7.  .  =  4,-4;  1^-    -=±2,  ±-V5; 

^  =  ^''=''-  .  =  :F3,TiV5.     " 

8.  a;  =  2,  3;  5 
2/  =  3,  2.  13.    a;  =  8,  4; 

20  V  =  4,  8. 

9.  a:  =  -2,  ^;  ^ 

8  14.    a;  =±2,  ±14; 

2/  =  -^'n*  2/  =  T3,  :f5. 


ANSWERS.  33 


10.    flj  =  -,  -;  18.    x  =  -.-: 
2   9  4    7 

=  1   1  -_1    _1 

^~9'2*  ^~      7'       4* 


16.  x  =  2a,  —a  —  b;  19.   x=S,  4,  — 4±V—  11  ; 
y  =  a  +  b,  -2a.  2/ =  4,  3,  -4:f  V^Hl. 

17.  x  =  -,  -3;  2^^    a;  =  ±3,  ±2; 


2/=±i,  ±V3. 

2^  =  :f  2,  1:3. 

21. 

05=1,  -3,  1  ±V-2;  2/ 

=  -2 

3,  1,  lq:V-2. 

22. 

.=  l,2,^±^-^^,= 

~3±V-55 
'       ^'              2 

23. 

a;  =  2  a,  —  a  ; 

32. 

,_2,i,    3±V-19 

y  =  a,  —  2  a. 

2         ^ 

24. 

ic  =  4,9; 

3TV-19. 

y  =  9,  4. 

2 

25. 

x=  a  —b,  b  —  a; 

33. 

05-2,--, 

y  =  a-\-b,  2a. 

2/  =  -l,  2. 

26. 

05=2,3; 

34. 

ic  =  4,  i; 

y  =  3,  2. 

9' 

27. 

a;  =  6  ±  a  ; 

0    22 

0    59 

2/  =  — 6  ±  a. 

2=3,  — 

28. 

a;  =  4,  16,  -12±V58; 

9 

2/=5,    -7,   -IqiV^S. 

35. 

^1     ^59 

29. 

0^  =  9,  5^2_g^; 

4 
31 

.     20 

2/  =  t3,  ±^- 

2/ =  4, 

8 

^        '  117 

a5=±2,  ±iv7; 

36. 

05  =  3,  -7; 

30. 

2/  =  -l,  -21. 

2/=±5,  iFyV7- 

37. 

.  =  3,2,-^^/^^^ 

31. 

a;=2a  — 6,  a-26; 
2/  =  a  — 26,  2a  — b. 

,=  2,3,-^^V309 

34  ALGEBRA. 

Art.  278;  pages  243  to  245. 

1.  9,  5.  3.    900  square  rods,  400  square  rods. 

2.  13,  6.  4.    3,  4. 

5.  Length,  10  rods;  breadth,  6  rods.  6.    7,  4. 

7.  Duck,  $1.75;  turkey,  $2.25.  q        _ig 

9.    -  or  

8.  21  or  12.  5         24 

10.  Length,  150  feet;  breadth,  100  feet. 

11.  Length,  16  rods  ;  breadth,  10  rods  ;  or,  length,  13i  rods  ; 

breadth,  12  rods. 

12.  Rate  of  the  boatman,  4  miles  an  hour;  of  the  stream, 

2  miles  an  hour. 

13.  A,  40  acres  at  $8  ;  B,  64  acres  at  $5.  14.    7,  5. 
15.    5,  2.         16.    Hind-wheel,  4  yards  ;  fore-wheel,  3  yards. 

17.  First  rate,  7  per  cent ;  second,  6  per  cent. 

18.  Length,  16  yards;  width,  2  yards. 

Art.  281 ;  page  247. 

9.  x^-9x  =  -20.  15.    6^2 +  31  a;  =  -35. 

10.  x'  +  2x  =  S.  16.    3ic2-f  17a;  =  0. 

11.  5a;2-12a;  =  9.  17.    ;^-2ax -bx  =  -a'-ab+2b^. 

12.  3a?2_2a;=i33. 

13.  12a;2-17a;  =  -6. 

14.  21ic2-|-44a;  =  32. 

Art.  282 ;  page  249. 

6.  0,  i.         9.    2,-4,-5.      16.    -1,  2,  ±3,  ±4. 

17  1     1  ±^^-S 

17.     -1,  ^-— 

3    _3±3V-3 


18. 

X-  —  2  mx  =m'^  —  m^. 

19. 

X'-4:X  =  -1. 

20. 

4  a^  —  4  mx  =  n  —  m?. 

10. 

0,  ±3.    12.    0,-4. 

13. 

14. 

.  ^    a±Va2  +  4& 

±a,             2 
""U'       3 

18. 


19.    ±1 


2'  4        

±1  ±V^^ 


ANSWERS. 


35 


20.  -1,±V^. 

21.  1,  ±3.      22. 


23.    a,  ±aV-l. 


5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 


25. 


±1.      24. 


^   ±1 


Art.  283 ;  pages  251  to  253. 


(x-{-6){x-10). 
(x  +  8)(x-\-5). 
{x-2){x-d). 
(2x-{-S){x-6). 
{4:X-S){x-3). 
{x-^7){6x-^l). 
(13+ a;)  (3 -a;). 
(1  + 2a;)  (2 -3a;). 


13. 
14. 
15. 
16. 
17. 
18. 


(a;  +  2+V3)(a^+2-V3). 

{Sx-l-{-^o)(Sx-l-^5). 

(4:X-S){'2x-3). 

{2-\-x){S-2x). 

{l-h2x){5-Gx). 

(3x-2  + V3)(3a;-2-V3) 

19.  {5-{-2x)(l-4x). 

20.  (10x-3)(a;-2). 

21.  (2a;  +  3)(8a;-|-5). 

22.  (V17  +  4  +  a;)(V17-4-a;). 

23.  (5  +  12a;)(3-2a;). 

24.  (5a;-2  +  V6)(5a;-2-V6). 

(6a;  4- 5a)  (.T- 3a).  27.    {ix  +  5y)(Sx -2y). 

(4a;  +  5m)  (5a; -h 4m).      28.    (7a;  — 3m7i)(3a;— 7m7i). 

29.  (x^Sy-2)(x-2y-\-3). 

30.  (a;-h22/-h2)(a;  +  2/-hl). 

31.  {S-\-2y-x)(2-Sy  +  x). 

32.  (x-2y-4:z)(x-Sy-z). 

33.  (2x-\-y-l)(x-y-{-2). 

34.  (a  +  6  +  3)(3a-f-6-4). 
38.    (x'-\.x  +  \)(xP-x+l). 


39. 

40. 
41. 
42. 
43. 


(a^-\-3x  +  l)(a^-3x+l). 
{2a'-h2ab-  b^)  {2a^-2ah-  b^), 
(m^  -|-  4  mn  -j-  n^)  (m^  —  4  mn  -\-  n^) . 
(l_|.36_262)(i_35_252). 

(^x^  ^4a:y  +  2y'){a^  -4.xy  -\-2f). 


36 


ALGEBRA. 


44.  (2a2  4-2a  +  3)(2a2-2a  +  3). 

45.  (2m2  +  2m-5)(2m=^-2m-5). 

46.  {d'-^ax^S-a^){a'-ax-^S-x''). 

47.  (x^'^Sx^2  +  d){x^-Sx-^2-^9). 

48.  (2a2  +  a&  +  4&2)(2a2-a6  +  462). 

49.  (4 aj2  4-  5 7,ia;  -Sm-)  {4a^  -  5mx -3 m^) . 

50.  (3ic2  +  3a;V2  +  2)(3.T2_3^^2  +  2). 

51.  (3a2  +  4am  +  5m2)(3a2-.4«m  +  5m"). 

52.  (2  +  2w-7n2)(2-27i-7n2). 

53.  (4:X^-hSxy-5y')(4x'^-Sxy-6y'^). 


Art.  284 ;  page  253. 


1.    ±V2± 


2. 


4. 


±V2±V-2 


2. 

±1±V2. 

5.    ±1±VS 

;. 

3. 

±  V3  ±  V- 
2 

-1^ 
Art. 

6.    ±^1^± 
4 

295  ;  pages  259,  260. 

V- 

2^ 

3. 

a;<4. 

7. 

x<a-\-h. 

11. 

8. 

4. 

a;<l. 

8. 

x>2,      y>4. 

12. 

19. 

5. 

^>lf 

9. 

a;<24,    y>S. 

13. 

32  or  33. 

6. 

a;<2a. 

10. 

Art. 

a;>  5  and  <  15. 
325;  pages  274,  275. 

3. 

'■  '-s- 

7. 

12.             9.    5^3. 

11. 

.    2,-3, 

4. 

■•■  «■  p- 

8. 

14 V3.     10.    3,-1. 

12. 

-!■ 

13. 

a;=±6, 

15.    25,   20. 

18. 

8,  18. 

14. 

2/=±10. 
x  =  ±  a'b, 

16.    23,  27. 

19. 

26,   14. 

y  =  ±  ab^. 

17.    9,  3. 

24. 

17,   12. 

ANSWERS.  37 

^5.    12,  8.      26.   A,  $105;  B,  $189  ;  C,  $270.       27.    8:7. 
28.    First,  1:2;  second,  2  :  1.  29.    ^^,  ^^. 

Art.  335 ;  pages  278,  279. 


3. 

63. 

8. 

21 
lO' 

13 

.    10  inches. 

4. 

.=i'- 

9. 

±|V3. 

14 

.    6  inches. 

5. 

6. 

10. 

5xand-l 

X 

16 

.    10. 

6. 

70. 

11. 

9. 

16 

.    8. 

7. 

14. 

12. 

3(V2—  1)  inches. 

17 

14 

4  —  5a; 

Art.  340;  page  281. 

2. 
3. 

S  =  540. 
Z  =  -69, 

6.    i  =  f, 

9. 

>S  =  -620 

. 

s  =  o. 

10. 

1  =  5, 

4. 

Z=57, 

S=17. 

/S'  =  552. 

'■  '=!■ 

11. 

J      137 
^"  15' 

5. 

;  =  -i45 

>S'  =  -217 

5. 

-!•■ 

«      917 
^^=15- 

Art.  341 ;  pages  283  to  285. 

4. 

a  =  3, 
S=  741. 

'•  "-h 

9. 

a=  5, 
d  =  -S. 

5. 

a=li, 

Z  =  -12^ 

1 

-. 

l=-li. 

10. 

71=16, 
I  =-4:3. 

6. 

'=h 

8.    d  =  -2i, 

11. 

n=18, 

;S  =  39. 

n=13. 

/S'  =  411. 

38  ALGEBRA. 

12.  a  =  3,  15.   d  =  -l,  18.   n  =  14, 

^  =  -^^-  n=.U.^  '  =  -151- 

13.  a  =  i  -I;  16.    d  =  5.  20.    d      '-« 


21.    d  =  g('^-«») 


»=10,  12.  ;  =  ^. 

14.    a  =  -l,  ,,  ^*       ,  M"-l) 

2'  17.  a  =  4,  —  5; 

d=2.  ?i  =  52,  43.  ^  = 

22^    ^^2S-n(n-l)d 


2S-an 


2n 


24.    a  =  Z-(n-l)cZ, 


OQ  l  —  a-^d  OK  2S  —  nl 

23.    n  = ! — ,  25.  a  = , 

d  n 

j^^{l  +  a)(l-a  +  d)  ^^^(nl-S) 
2d               '  n{n-l)' 

26     ;^-^^  v'^^^+(^^-^?- 
It 


07     ^  ^'-a'  OQ  2Z+d±V(2/+d)2-8d^ 

2/S  — a  — Z  2d 


2S  ^^dq:  V(2Z  +  d)^-8d>S 

a  +  Z*  2  * 


Art.  344 ;  page  286. 

1.  d  =  l.  3.    d  =  -\\^     5.    d  =  -i.       7.    2a6. 

Q  0 

2.  d  =  -i.       4.   d  =  i  6.    -i.  8.   ^!±^'. 

2  7  15  a^-V" 

Art.  345 ;  pages  287,  288. 

3.  2500.  6.    11.  g  23a^ 

4.  -43.  7.    Z=10m-27n,         '      3  * 

5.  4,11,18,25.  >S'=55m-135n.     9.  62750. 

10.    2,  6,   10,   14;  or,    -2,    -6,    -10,    -14.  11.    22. 


ANSWERS.  39 

10  A        ^    o    K    Q  59    165         11        7 

12.  -4,  -1,  2,  5,  8;    or, —,—,—,-—,  --. 

14      7      14         7         2 

13.  After  9  days,  at  a  distance  of  90  leagues. 

14.  4117^  feet.  15.  3,  7,  11  ;  or,  4|,  7^,  10^. 
16.  8.           17.  $2950.  18.  852. 

Art.  349 ;  pages  290,  291. 

3.  Z  =  256,       7  ^^JL_  10  Z=— L, 
>S'=511.            2048  324 

4.  ,^_6i,         ^^2047.         ^^IL. 


243  2048  162 

Z=192, 


^^2059       g  ?==-I?2,     11.  Z=192, 
243  64 


6.  Z  =  2048,  ^^   1261 

>S'=:1638.  192* 

6.  Z  =  --i-,  9.  1  =  —,                 12.  Z  = ?-, 

256  32  768 

^^341  ^  =  yi.  S  =  -—' 

256*  32  '  256* 


Art.  350;  pages  292,  293. 

3.  a  =  i,         7.  a=-,     11.  a=-l, 
2  3 


?i=  8. 
12.  ^^a+(r-l)5 


r 


^^1023  ?  =  -l_ 

2  *  6561 

4.  a  =  -?,        8.  r  =  -^, 

2  4        _^-a 

^=48.  S  =  ^^  '  "  S-l' 

1024* 

5.  r  =  3,  -3;      9.  r  =  i,      H.  a=rM--l)^. 

/S'=  2186,  1094.        ^ 

?i=  7 

6.  n  =  5,         10.  l  =  - 


15.  a  =  -?-, 
243         r"-^ 


>S'=121. 


2  '     ^^  l(r--l) 
n  =  6.  r"  \r  — 1) 


40  ALGEBRA. 

16.  „=(!lrl)^,  i^r'-Hr-i)S^ 

r"  —  I  r""  —  \ 

17.  r  =  -v-'    ^=irT7^ — ;r-i7 — 

Art.  351 ;  page  294. 

2.  4.  4.   -i  6.  5.  8.   -1^. 

4  4  19 

3.  1  5.    -1^.  7.   ?5.  9.       ^ 


3  4  11  a2  +  aj2 


Art.  352  ;  page  295. 

^   ii.     5.    i^.     6. 
11       '      27  15  165  825  1100 


2.    £.     3.   il.     4.   ii.     5.    i^.     6.   ^.     7.     2^^ 


Art.  355 ;  pages  295,  296. 

1     2     -     -     —     —     — 
'    3'    9'    27'    81'    243* 

o      4-5     9^_  27     81         243 

*  2'    2'  "^  2  '     2  '         2  * 

3.    -6,   -18,   -54,   -162,   -486,   -1458. 

'^'i'    8'  "^16'    32'       64'    128'       256* 
5.    ±4,   -8,    ±16,   -32,    ±64. 

4'    16'       64'    256  ^  6 

Art.  356;  pages  296,  297. 
2.    3.  3.    5,  10,  20,  40;  or,  -15,  30,  -60,  120. 

4.    5,  15,  45  ;  or,  40,  -  20,  10.         5.    ±4.         6.    $64. 

.y     QiAA^    ^      Q     o    .    o    1^  810         540     360         240 

7.    3100  feet.     8.   2,  4,  8,  16  ;  or, — ,  —  • ,   , 

'    '    '       '      '  13  13       13  13 

9     ._§L.  ^^-    3,9,27.  12.    1,  2,  4. 

8192*  11.    2,4,8;  or,  -  2,  4,  -8. 


ANSWERS. 

Art.  361 ;  page  300. 

3. 

3                A               1                 K             5               ft 

—         4. 0. b. 

74                      78                      4 

3 
4' 

7.    - 

8. 

48      24     16     12      48       8       48 
125'    65'    45'    35'    145'    25'    155 

11. 

15. 

9. 

40         20         40 

~r7'  "T'  ~n'      ''• 

12. 

a'  -  b' 
d'-\-b' 

10. 

21         7         21          21^ 
7,   -21,    --,        -,        ^,        ^^ 

13. 

a6                   j4      ab(m-^l) 
7ia  —  nb  —  a-i-2b       *       6m +  2  a  — 6 

15. 

3 
19* 

41 


3 
142* 


Art.  365 ;  pages  303,  304. 

2.  c^  +  4c2(r^  +  6c*d~^H-4c'cr^ -f  «^~'^. 

3.  m~^  —  5m  V  +  10?/r^n*  —  lOm-^71^  +  5m~^n'  —  n^\ 

4.  oi^y^ —  dxy~^ -\-21  x~^y —  27x~^y^. 

5.  .-r""  +  lOx^'^y'' -f  40ar^ V  +  SOar^-^"  +  80a;V*  +  S2y^'\ 

6.  a'^-{-12a^x^  +  64:a^x-{-l08a'x'-^8\x^. 

7.  m*n-^  —  6m^n~^  +  10m%~2  —  lOm^n"^  +  5  m^n  —  m^nK 

8.  xy-^-{-3y-'-hSx^-^x-Y 

9.  7M*  —  2  m^rr'  +  -  m*?i^ 7n^n^  H n^^. 

2  2  16 

10.  ah~^  -5ah-^  +  10a''6"L  I0a~^6^  +  5a'^6  -  a'^di 

11.  a«-12a'^'4-54a"'^'-108a^4-81ai 

12.  16a;V^  +  16a;V^  +  ^^  +  aj"V  +  -^^"V* 

16 

13.  a-^2  -  2 a-%^  +  -a-«a;  _  ^^a-^ajt  +  -^a-^ar' 

3  27  27 

81^    ""   +729     ' 


42  ALGEBRA. 

14.  a^4-15a;"'^V^+  dOx^y~^ -{-270 xhj~^ -\- 4:05 x^y~'^  +  2^3 y-\ 

15 .  I  a^b^x~ ^  -  -  b^x-s  +  6  a-^b~'^x^  -  8 a.  %~^x. 
8  2 

16.  81  a-%^  -  108a-26  +  54a-i  -  126-i  +  ab~\ 

17.  a^ft  3 _|_  12 a25-2  +  60a6-i  +  160  +  240a-i6 

+  192a-252_,_  34^-3^3^ 

18.  l-4.x  +  2x'^^x'-bx^-So^  +  2:^-\r^x''  -^-a?. 

19.  a^  +  4a;7— 2a;«-20af +  a;^+40aj3-8a^-32a;4-16. 

20.  1  +8a;4-20ar'4-8a.-3-26ic^-8a^+20a;^-8a;^  +  a^. 

21.  l-5a;+15.^2-30a^  +  45a;4-51a^  +  45a;^-30a;^ 

H-15a^-5aj9  +  a.'^°. 


Art.  367 ;  page 

305. 

2. 

462  aV. 

6.  84a-36'. 

9. 

IbmOx-^y^. 

3. 
4. 
5. 

252  m^ 
-  792 c^cf. 
1001  a«. 

7.  715  a;". 

8.  -^a-'b\ 

16 

10. 
11. 

-4455a'^V^ 
126720. 

Art.  378 ;  pages  311,  312. 

2.  l-2x  +  2a;2_2aj3_^2a;^ 

3.  2H-lla;  +  33a;2 4-99x3  +  2970;*+. •. 

4.  3 -19a;2_^  95^.4  _  475^6  _j_2375a^ 

5.  ^x  +  ^o^  +  -x^  +  -x'  +  —x'  +  "' 
3        9  27         81  243 

6.  l-2a;  +  2a;3_2a;4  4.2a;6 

7.  x-Q(?-2;K?-bx^-\2a^ 

8.  2-a;  +  3a;2_^^3^4 

9.  l-2a;  +  5ar'-16ar^  +  47a^ 

10.    2-7a;+28a;2_9i^^322a;4 


ANSWERS.  43 


ie     2    _2  ,  8   _i  .  32^128^  ,512   «, 
3  9  27       81         243 

16.  x-^  +  S-h2x-6x'-Ua? 

17.  x-^-x-^-2x+2x^-4a^-\-'" 

18.  ^x-'-lx-'--x''-\-  —  -\-—x  +  "' 
2  4  8  16      32 


Art.  379 ;  page  314. 

2.  \^x--x'-\--a^-^x*+'-' 

2  2  8 

3.  X-^-x-^-x^-'^x^-'-^x^-.. 

2        8  16  128 

4.  \-x-{-x^  +  a^-h-x^-{-'" 

2  8         16  128 

6.  \-lx-^-x?-^x''-^x^->>> 

3  9         81  243 

7.  \J^\xJr^-^-—^^—x'^-- 

3        9         81  243 


Art.  381 ;  page  316. 


3. 

4. 
6. 

^       1       ^ 

2aj  +  5      2a;-5 
3           5 

X     3x4-5 
5          1              1 

6.    -?  +  ^-4-^- 
a;      3a;  4-1      2a; -5 

-      3a  2a 


a;4-  a      a;  —  4a 

8.   -^  +  ^. 
2a;     a;4-3      a;-3  3  +  4a;     3-a; 


44  ALGEBRA. 


9.    -i-+      2  3 


10. 


1  1  3.3 


x-{-2      x  —  2      x-\-l      x—1 


11.         1+V2        I        1-V2 


2a;-5-hV2      2a;-5-V2 


Art.  383 ;  page  317. 

23  3.    -i-+       ^ 


oj-f  5      (aJ  +  5)2  07-2      (ic- 2)2  ■   (£c-2) 

4         3  6  1 


a; -1-1      {x-\-iy      (0^4- 1)^ 
2  4  3 


3a;  +  2      (3a;  +  2)"      (3x+2)3 

1 4 

5  (5a; -2)      5  (5  a; -2)3* 


7,   -^+        1  1  1 


ic  +  i    {x-hiy    {x  +  iy    (x-^iy 

Q        2  4^3  1 


9. 


a;- 1       (a;- 1)2      (a;-!)^      (x  -  ly 

1 27 27 

2  (2a; -3)      2(2a;-3)«      (2a; -3)^* 


Art.  384 ;  page  319. 
2.    ? § 5_.  3     5_2 


4:     ,i;  +  2       (a;  +  2)''  a;      re«      ai+l       (a;  +  l)' 

4.   -J '—+  ' 


x-2      2  (2a; -3)      2  (2a; -3)' 

.5.  i+^+-i,+    1 


a;      a;-l      a;  -  2      (a;-2)^ 

6.  1-1  +  4-  * 


a;      ar^      x^     a;-f-5 
7.   5-1  +  1==.     ^ 


X      x^     s^     x-\-l      (a;  +  l)' 


ANSWERS.  45 

Art.  385 ;  page  320. 


1.    2aj-5 !^+       ^ 


2a;— 5      2a; +  1 
2     3_^     1  5  18 


a; +  2       (.^  +  2)2      (a; +  2)' 
3.    5a;2  4.1_l+  3         1 


X      x^      X'      x-ir\ 
4.    3a; -2 ?— _^ ? — -+     ^      4.        ^ 


a;-f-l      (a;+l)-      a;  -  1      (a;-!)^ 


6.    2a;2_7_2_^i  5 


X      x^      x—1 
Art.  387 ;  page  322. 

2.  x=^-y-{-^y^--l-f--^y^^... 

^^      27^       243^       2187^ 

3.  a;  =  22/4-6/4-Y/  +  9S2/'  +  - 

4.  a;  =  (2/-l)- I  (y-l)2  +  i(2^_l)3_  1(2,-1)44..., 

5.  a;  =  2/  +  2/'  +  2y'  +  5/+... 

6.  ^  =  22/  +  |2/^  +  ^2/3^464^.^... 


7^1    5  ,  ,  29  ,   199  . 


2/*  + 


8.  .  =  ,  +  1/+^^  +  !^,.  +  ... 


Art.  392  ;  page  327. 

7.  a^-\a-^-x--ar'^x'-~a-ix? La-^a;*-... 

2      8       16      128 

8.  l-la;  +  ?a;^-iia;3  +  l^^._.,, 

3    9    81    243 


46  ALGEBRA. 

9.    a-^  +  3a-*x  +  6a-^x^  -j-lOa-^ar^  -i-loa'^  X*  -\-  '" 

10.  c"^  -  c-^d  +  c~^d-  -  G-^d^  +  c-~^'d' 

11.  x~^  —  2x^y  —  x^y^ x'^y^ x^y^  —  ••• 

o  o 

2  8  16  128 

13.  a-s  -  a'h-'  +  -a-'^"b''  --a-"^ b-'  +  —  a-'b  ' 

4  2  16 

14.  a;3  +  dx-^ab  -  -x-'a'b'  +  -x^'a'b'  - Maj-i^a^^^  +  ... 

2  2  8 

16.    l-10xy-'  +  80x'y-'-^x'y-'-^^^x'y-'-'- 

o  o 

16.  a*+12a^2/-2+90a^2/''  +  540a^?/  «  +  2835a«2/~«4---- 

17.  12Sa^  +  lV2a'x-^  +  S6a^x~^  -h^ax-'-{-■P^a-^x~^- 

8  256 

18.  m  +  3m*n^  +  -m^n^  +  5^m^n'+^m'^V+- 

^  Z  o 


Art.  393 ;  page  328. 

2.  JLa-^a;^       6.    -^^Ma-^^^t.       10.    SOOSn^'c-'^. 
2048  256 

«^.     II  »       44     JL4      „  .,  308   -34 

3.  -364m".  7. x^  y-\  11. —a   ^  x-\ 

6561  3 

128  8192         ^  ^ 

5.    --A_a-^a^.    9.   -ll-a,-'?'a;i«. 
1024  256 


Art.  394 ;  page  329. 

2.  3.16228.  4.    2.08008.  6.    2.03055 

3.  9.94988.  6.    2.97182.  7.    1.94729 


ANSWERS.  47 


Art.  407 ; 

page  333. 

2. 

1.3222. 

7. 

1.9912. 

12.  2.1303. 

17.  3.0545. 

3. 

1.7993. 

8. 

2.0212. 

13.  2.2252. 

18.  3.7114. 

4. 

1.7481. 

9. 

2.0491. 

14.  2.1673. 

19.  3.8484. 

5. 

1.9242. 

10. 

2.1582. 

15.  2.5741. 

20.  4.1585. 

6. 

1.6532. 

11. 

2.3343. 

16.  2.5353. 

21.  4.1915. 

Art.  409 ; 

page  334. 

2. 

.3680. 

6. 

1.5441. 

8.  .2252. 

11. 

.8539. 

3. 

.1549. 

6. 

.1182. 

9.  2.2431. 

12. 

.7660. 

4. 

.5229. 

7. 

2.0970. 
Art.  412 ; 

10.  1.0458. 
page  335. 

13. 

.7360. 

3. 

.2863. 

9. 

4.5844. 

15.  .1165. 

22. 

.2601. 

4. 

2.7090. 

10. 

3.2620. 

16.  .3860. 

23. 

.6884. 

5. 

4.2255. 

11. 

.9801. 

17.  .2212. 

24. 

.1840. 

6. 

.1398. 

12. 

.4225. 

18.  .1750. 

25. 

.2215. 

7. 

.7194. 

13. 

.1590. 

20.  2.6145. 

26. 

.2494. 

8. 

.6611. 

14. 

.0430. 
Art.  414 ; 

21.  .1678. 
page  337. 

27. 

.1449. 

2. 

.2552. 

7.  7.7; 

323-10.    12. 

2.4804 

3. 

.3522. 

8.  6.4983-10.    13. 

8.7905 

-10. 

4. 

9.2922- 

-10. 

9.  3.8663.       14. 

6.3588 

5. 

8.6811- 

-10. 

10.  .60 

74.        15. 

.1964. 

6. 

1.5841. 

11.  9.6511-10.    16. 

.1688. 

Art.  420; 

page  341. 

7. 

9.8878- 

■10. 

11.  1.3028.       15. 

0.7144. 

8. 

3.0237. 

12.  4.9659.       16. 

3.0155, 

9. 

0.5177. 

13.  9.6055-10.   17. 

8.9379 

-10. 

10. 

8.7164- 

•10. 

14.  7.8560-10.   18. 

9.0610 

-10. 

48 

ALGEBRA. 

Art.  421 ;  page  343. 

6. 

1.646. 

10. 

.003318.        13. 

.2079. 

16.    63329. 

7. 

8886. 

11. 

10221.            14. 

44.48. 

17.    .01301 

9. 

.01461. 

12. 

9.492.            15. 

.001109.      18.    502.9. 

Art.  426 ;  pages  346  to  348. 

1. 

8.454. 

19. 

-1.184. 

39. 

.6443. 

2. 

10.73. 

20. 

.000007038. 

40. 

.5010. 

3. 

-  2202. 

21. 

2.924. 

41. 

1.062. 

4. 

.2179. 

22. 

.9146. 

42. 

-.9102. 

5. 

.01157. 

23. 

4.638. 

43. 

1.093. 

6. 

-.7032. 

24. 

.0000639. 

44. 

.7035. 

7. 

7.672. 

25. 

1.414. 

45. 

.5807. 

8. 

.6688. 

26. 

1.495. 

46. 

-.6313. 

9. 

-3.908. 

27. 

-1.246. 

47. 

24.62. 

10. 

1782. 

28. 

.6553. 

48. 

.2979. 

11. 

.3500. 

29. 

.2846. 

49. 

98.50. 

12. 

-  .4748. 

30. 

2.372. 

50. 

1.660. 

13. 

.4127. 

31. 

-.5142. 

51. 

3.076. 

14. 

-4.671. 

32. 

.1588. 

52. 

.8678. 

15. 

.2415. 

35. 

5.883. 

53. 

1.134. 

16. 

-  .0725. 

36. 

.7885. 

54. 

.5881. 

17. 

13587. 

37. 

1.195.    ' 

55. 

1.805. 

18. 

.006415. 

38. 

.6803. 

56. 
57. 

.003229. 
.03344. 

Art.  427 ;  page  349. 

3. 

.4581. 

4. 

.1853 

5.    - 

.4949. 

6.    -.2601. 

7. 
8. 

m  log  h  -\-n 
log  a 
log  a 

.l0g( 

-• 

9 
11 

.    3,  _i.          10.    -3. 

log /-log  a  ^ 

log  n  —  log 


m 


logr 


ANSWERS. 


49 


12.  ^^log[0--l)^  +  «]-loga, 


13.  n  = 


los:  I  —  losf  a 


+  1. 


log(>S-a)-log(6'-0 

logr 

16.  3.4598.      18.  -3.467.     20.  .9395. 

17.  -.1386.     19.  11.193.      21.  -1.8204. 


3.  4.479. 


Art.  437 ;  page  353. 
4.  7.19.     5.  -1.07.     6.  -2.4576. 


1.  $2853.75. 
6.  14.198. 


Art.  439 ;  pages  356,  357. 
2.  $702.86. 
6.  16.01. 


3.  5^. 


4.  4. 

7.  $647.14. 


Art.  443;  page  359. 

I.  $2076.40.   3.  $2959.18.    4.  $277.  6.  $576.50. 

Art.  452  ;  pages  363  to  365. 

3.  7893600.      5.  126.         7.  3838380. 

4.  5040.         6.  15120.        8.  31824. 

9.  360;  120;  720;  1956. 

10.  134596.      14.  15840.       17.  10584000. 

II.  125970.      15.  121030.      18.  3303300. 
12.  4536.        16.  10080.       19.  720. 


Art.  466 ;  pages  375,  376. 

1.    1  -\ ;  5th  convergent,  — -• 

2+  1+3+1+2  ^      '  14 

o        1       1       1       1       1     1       r.i  4.    15 

2. ;  5tn  convergent,  — -• 

1+  3+1+3+1+3  ^         19 


50  ALGEBRA. 


3.   3  H :  5th  convergent,  — 

1+  1+  1+  1+  3+2+2'  ^5 

-11111111.,,  ^8 

4. ;  otn  convergent,  — 

1+  2+  1+  2+1+2+1+2  ^       '  11 

6.  2  H :;—  ;; I  5th  convergent,  — 

3+  2+  1+3+2+1+2  ^      '  37 

o.    1  H ;  5th  convergent,  -• 

1+1+1+1+1+1+1+1+2  ^      '5 

,y     1    ,     1      1      1      1      1      1     1      . .,  .24 

7.  1  + :  5th  convergent,  — 

3+  1+  3+1+3+1+3  ""         19 


«      1      1      1      1      1    1     ..,                     .68 
o.    —  - —  - — ;  5th  convergent, 

2+  3+  4+  5+  6+7  ^         157 

Q       O    L    1  1  .I^T,  *     161  1  1 

y.    2  H ;  4th  convergent, ;  , 

4+  4  +  -  ^       '72      27144    21960 

10.  1  H ;  4th  convergent,  -  ;  — ,  — 

1+  2  +  -  *=      '  4'  60   44 

11.  3  H ;  4th  convergent,  ;  , « 

3+6  +  -  ^       '    60  '  26340    22740 

12.  2  +  —  —  J: — ;  4th  convergent,  - ;  — ,  — • 

1+  1+  1+4+-'  ^         3'  51' 42 

-q     -34-V15  IK     3+V5 

13. 15.  — ^— . 

14.  -2  +  2V2.  16.    -H-2V6. 

17.    3  +—  — —;  4th  convergent,  — • 

7+  15+  1+-  "^         113 

-g       111111       1.76.        1 1__ 

*    2+  3+  3+  3+  1+  1+  7+-  '  175  '  262325'  231700* 

19     24-i- J--i--i--i- J-_L__L_.   19?. 

1+ 2+  1+  1+ 4+ 1+  1+  19  +  -'    71' 

1  1 

103589'  98548* 

20.    3  +- J-  -J— ;  5th  convergent,  — • 
1+  5+"'  41 


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